UZBEK MATHEMATICAL JOURNAL

UZBEKISTAN ACADEMY OF SCIENCESV.I.ROMANOVSKIY INSTITUTE OF MATHEMATICS

UZBEKMATHEMATICAL

JOURNAL

Journal was founded in 1957. Until 1991 it was named by "Izv. Akad.Nauk UzSSR, Ser. Fiz.-Mat. Nauk". Since 1991 it is known as "Uzbek

Mathematical Journal". It has 4 issues annually.

1. 2018

TASHKENT - 2018

Editorial BoardEditor in ChiefSh.A. Ayupov – (Functional Analysis, Algebra), V.I.Romanovskiy

Institute of Mathematics, Uzbekistan Academy ofSciences (Uzbekistan), sh−[email protected]

Deputy Editor in ChiefU.A.Rozikov – (Functional analysis, mathematical physics),

V.I.Romanovskiy Institute of Mathematics,Uzbekistan Academy of Sciences (Uzbekistan),[email protected]

Managing editor

K.J.Khalliev – Managing editor of the Uzbek MathematicalJournal (Uzbekistan), [email protected]

Editors

A.Azamov – (Dynamical Systems, Game Theory, DifferentialEquations) V.I.Romanovskiy Institute ofMathematics, Uzbekistan Academy of Sciences(Uzbekistan)

Sh.A.Alimov – (Mathematical Analysis, DifferentialEquations,Mathematical Physics) NationalUniversity of Uzbekistan (Uzbekistan)

R.R.Ashurov – (Mathematical Analysis, Differential Equations,Mathematical Physics) V.I.RomanovskiyInstitute of Mathematics, Uzbekistan Academyof Sciences (Uzbekistan)

Aernout van Enter – (Probability and mathematical physics)University of Groningen (The Netherlands)

Y.Kh.Eshkabilov – (Functional Analysis), National University ofUzbekistan (Uzbekistan)

Sh.K.Farmonov – (Probability thory and Mathematical Statistics),V.I.Romanovskiy Institute of Mathematics,Uzbekistan Academy of Sciences (Uzbekistan)

R.N.Ganikhodjaev – (Functional analysis), National University ofUzbekistan (Uzbekistan)

N.N.Ganikhodjaev – (Functional analysis),International Islamicuniversity (Malaysia)

A.R.Hayotov – (Computational mathematics), V.I.RomanovskiyInstitute of Mathematics, Uzbekistan Academy ofSciences (Uzbekistan)

3

Fumio Hiroshima – (Spectral and stochastic analysis, functionalintegration, Quantum field theory), Professor ofKyushu University (Japan)

E.T.Karimov – (Differential equations), V.I.RomanovskiyInstitute of Mathematics, Uzbekistan Academyof Sciences (Uzbekistan)

M.Ladra – (category theory: homological algebra,nonassociative rings and algebras), University ofSantiago de Compostella (Spain)

Lingmin Liao – (p-adic analysis, dynamical systems, numbertheory) University Paris-Est (France)

Arnaud Le Ny – (Probability and Statistics, Statistical Mechanics)University Paris-Est (France)

B.A.Omirov – (Algebra, Number theory), National University ofUzbekistan (Uzbekistan)

I.Rakhimov – (Algebra, Number theory), University of PutraMalaysia (Malaysia)

Lorenzo Ramero – (algebraic and arithmetic geometry, commutativerings and algebras), University of Lille (France)

A.S.Sadullaev – (Mathematical analysis), National University ofUzbekistan (Uzbekistan)

M.S.Salakhitdinov – (Differential equations), V.I.RomanovskiyInstitute of Mathematics, Uzbekistan Academyof Sciences (Uzbekistan)

F.A.Sukochev – (Functional analysis, Geometry), University ofSouth Wales (Australia)

J.O.Takhirov – (Differential Equations), V.I.RomanovskiyInstitute of Mathematics, Uzbekistan Academyof Sciences (Uzbekistan)

J.P.Tian – (applied mathematics including dynamicalsystems and partial differential equations andstochastic differential equations), New MexicoState University (USA)

V.I.Chilin – (Functional analysis), National University ofUzbekistan (Uzbekistan)

Kh.M.Shadimetov – (Computational Mathematics), TashkentInstitute of Railway Engineering (Uzbekistan)

O.Zaitov – (Topology), Secondary special educationdepartment of the Khorezm region (Uzbekistan)

Efim Zelmanov – (Agebra, Jordan Algebras, Infinite DiscreteGroups, Profinite Groups), UC San Diego (USA)

Postal Address: Mirzo Ulugbek str., 81, Tashkent 100170, Uzbekistan

4 Ashurov R., Butaev A.

Uzbek MathematicalJournal, 2018, No 1, pp.4-26

On pointwise convergence of continuous wavelettransforms

Ashurov R., Butaev A.

Abstract In this paper we give a general information on wavelettransforms and present a short survey on convergence of continuouswavelet transforms. First we discuss how wavelets were discovered. Thenwe compare Fourier and Short-time Fourier transforms with Continuouswavelet transforms. After this introduction we consider wavelets withsufficient decay at infinity and investigate convergence of continuous wavelettransforms. To study wavelet transforms, associated with general (withoutany decay or smoothness conditions) spherically symmetric wavelets, weintroduce the Riesz summation method. We also define and study theclassical and generalized localization principles of spherically symmetricwavelet transforms of functions and distributions. Finally we introduce anew class of multidimensional wavelets which has a very good convergenceproperties. The main results of this report have been published in the papers[1]-[6]

Keywords: Window function, Gaussian function, integrable functionMathematics Subject Classification (2010): 42C15, 42C40

1 Introduction

Wavelets are universal tool with very rich mathematical content and greatpotential for applications in various scientific fields. Although wavelets havea very short history; they appeared in 80 th of the last century, but waveletanalysis was used much before than it was given its name. An excellenthistorical account can be found in the books [7] and [8] and in the surveypaper of Meyer [9].

There are many related concepts have been known for decades, withother names. In 1909 Hungarian mathematician Alfred Haar in his PhDthesis has constructed an orthonormal system, such that decompositionsof any continuous function converge uniformly. Later the signal processingcommunity has been using this basis (moving average filter) without beingaware of Haar result.

On pointwise convergence of continuous ... 5

One of the most significant step towards to wavelets was done by D.Gabor. In 1946 he introduced so-called the windowed Fourier transform(WFT). The idea is to use a sliding window function g in order to localize theinfinite waves of exponential functions. However if we consider g(t− u)eitω

as wavelets, then we can see that they oscillate too much at high frequencies,and this leads to significant numerical instability in the computation of thecoefficients.

In 1982 so-called "wavelet revolution"started. In this year a Frenchengineer J. Morlet developed a new time-frequency analysis, using shortwaveforms at high frequencies. Such waveforms are simply obtained byscaling a single function called a "wavelet". His main wavelet ψ was definedas ψ(ξ) = ξ2e−ξ

2/2.In the same year, after collaboration with Morlet, a theoretical physicist

Alex Grossmenn together with T. Paul and I. Daubechies introduced thecontinuous wavelet transforms. In 1985 Yves Mayer, a mathematician,established the connection between continuous wavelet transforms andCalderon-Zygmund operators. Meyer’s mathematical knowledge allowedhim to give a mathematical foundation to wavelet theory.

In 1988 Ingrid Daubechies (a theoretical physicist) discovered hercelebrated compactly supported k-times differentiable scaling functions anddeveloped the application of these scaling functions to signal and imageprocessing.

Wavelet revolution lasted from 1982 to 1988. Of course, the history ofwavelet theory does not end in 1988. Much work has been done since then.Now we can say with confident, wavelet theory has become a scientific fieldof its own.

2 Fourier and Short-time Fourier transforms

When the function f depends only on one real variable t (t refers to time),i.e. f(t), then we call it a signal.

Let a signal f be defined in R = (−∞,∞). Then the integral∫ ∞

−∞|f(t)|2dt

is called the energy of the signal f . The class of all signals with finite energywe define by L2(R).


The Fourier transform of signals with finite energy is defined as

f(ω) =1√2π

∫ ∞

−∞f(t)e−itωdt (2.1)

and it is called the (continuous) spectrum of f . We can reconstruct thesignal f from its Fourier transform:

f(t) =1√2π

∫ ∞

−∞f(ω)eiωtdω. (2.2)

In signal analysis signals are defined in the time-domain, and the spectralinformation (i.e. the Fourier transform) of these signals is given in thefrequency-domain.

The Fourier transform is a very powerful mathematical tool inapplications. However, the formula (2.1) alone is not quite adequate formost applications, especially when we need a time-frequency analysis.

Indeed, to find the spectral information f(ω) at a single frequency ωwe must know information about the signal f both past and future, i.e. weshould use the quantity of f(t) from the infinite time interval (−∞,∞).Moreover, the formula (2.1) does not show how the frequency depends ontime. If we change the signal f in a small neighborhood of some time instant,then the entire spectrum will be effected.

The deficiency of the Fourier transform formula (2.1) in time-frequencyanalysis was first observed by D. Gabor in 1946, when he introduced hiswindowed Fourier transform (WFT). Gabor has chosen a Gaussian functionas the window function:

gα(t) =1

2√πα

e−t24α ,

where α > 0 is fixed.WFT of Gabor of a signal f is defined by

(Gαb f)(ω) =1√2π

∫ ∞

−∞e−iωt[f(t)gα(t− b)]dt, (2.3)

that is first we take a "part"of f inside the window around t = b, then weapply the Fourier transform. Note, here to extract the spectral informationfrom the signal f , we should know about f only around t = b.


Obviously, (Gαb f)(ω) is not the Fourier transform of f , but since∫∞−∞ gα(t)dt = 1, then if we sum WFT over all b, then we have f(ω):∫ ∞

−∞(Gαb f)(ω)db = f(ω), ω ∈ R,

i.e. the set Gαb f : b ∈ R of WFT of f decomposes the Fourier transformf of f exactly.

The Fourier transform of a Gaussian function is again the Gaussianfunction:

gα(ω) =1√2π

e−αω2.

Therefore by using the Parseval identity we have

(Gαb f)(ω) =1√2π

∫ ∞

−∞e−iωt[f(t)gα(t− b)]dt

=√π

αe−ibω

1√2π

∫ ∞

−∞eibξ[f(ξ)g1/4α(ξ − ω)]dξ. (2.4)

Hence the window Fourier transform of f with window function gα att = b coincides with the window inverse Fourier transform of f with windowfunction g1/4α at ξ = ω (with the exception of the of the multiplicative term√

παe

−ibω). Thus we have both the time and frequency localization.How to measure the "sizes"of these windows? To do this it is agreed

to introduce the windows "center"and "width"as follows (see, for example,[10]).

Definition 2.1. A nontrivial function w ∈ L2(R) is called a windowfunction if

tw(t) ∈ L2(R). (2.5)

The center t∗ and radius 4w of a window function w are defined to be

t∗ =1

||w||22

∫ ∞

−∞t|w(t)|2dt (2.6)

and

4w =1

||w||2

∫ ∞

−∞(t− t∗)2|w(t)|2dt

12

, (2.7)

respectively; and the width of the window function w is defined by 24w.


For example, for the function gα we have t∗ = 0 and 4gα =√α.

Therefore, for WFT we have the following time-window around b (see (2.3)):

[b−√α, b+

√α].

Since gα is also a window function with the center ω∗ = 0 and 4gα =(2√α)−1, then for WFT the frequency-window has the form (see (2.4)):[

ω − 12√α, ω +

12√α

].

The product of these two windows:

[b−√α, b+

√α]×

[ω − 1

2√α, ω +

12√α

]is called a rectangular time-frequency window. Observe that the width ofthe time-frequency window is unchanged for observing the spectrum at allfrequencies. This in fact restricts the application of the Gabor transform tostudy signals with unusually high and low frequencies.

Suppose we have any window function w instead of gα. If w is alsowindow function, then the transform is called Short-time Fourier transform(STFT). Then for any signal f its STFT (which we denote as Sbf) has theform:

(Sbf)(ω) =∫ ∞

−∞f(t)wb,ω(t)dt, (2.8)

where wb,ω(t) = e−itωw(t− b). If t∗ is a center and 4w is the width of thewindow function w, then to extract spectral information (Sbf)(ω) from thesignal we should know information about f only in the time-window aroundt∗ + b:

[t∗ + b−4w, t∗ + b+4w].

Let ω∗ and 4ω the center and width of the window function wrespectively. Then we have a time-frequency window:

[t∗ + b−4w, t∗ + b+4w]× [ω∗ + ω −4w, ω

∗ + ω +4w].

Again, the width of the time-frequency window remains unchanged forlocalizing signals with both high and low frequencies. However, thecontinuous wavelet transform provides a flexible time-frequency windowwhich automatically narrows when observing high-frequency phenomenaand widens when studying low- frequency environments.


3 Continuous wavelet transforms

Let ψ ∈ L2(R) satisfy the condition

Cψ = 2π∫|ω|−1|ψ(ω)|2dω <∞, (3.1)

which is called the admissibility condition for function ψ to be a wavelet. Ifψ ∈ L1(R), then ψ is continuous and (3.1) can only be satisfied if ψ(0) = 0,or ∫

ψ(t)dt = 0. (3.2)

Hence the function ψ in order to satisfy the admissibility condition shouldhave some oscillation. This is the reason that ψ is called a "wavelet".

We generate a doubly-indexed family of wavelets from ψ by first dilatingby the factor a and then translating by b:

ψa,b(t) = |a|−1/2ψ

(t− b

a

),

where a, b ∈ R, a 6= 0. The normalization has been chosen so that ||ψa,b||2 =||ψ||2 for all a, b. We will assume that ||ψ||2 = 1. The continuous wavelettransform of any signal f ∈ L2(R) by ψ is defined by

Wψf(a, b) =∫ψa,b(t)f(t)dt.

The difference between the wavelet and WFT lies in the shapes ofthe analyzing functions. The ψa,b(t) have time-widths adapted to theirfrequency: high frequency ψa,b are very narrow, while low frequency ψa,b

are much broader. Changing the parameter b allows us to move the timelocalization center: each ψa,b(t) is localized around t = b. As a result, thewavelet transform Wψf(a, b) is better able than the WFT to analyze thesignals.

Now let the wavelet ψ satisfy (2.5). Then if the center and radius of thewindow function ψ are given by t∗ and ∆ψ, respectively, the function ψa,b

is a window function with center at b+at∗ and radius equal to a∆ψ. Hence,the wavelet transform Wψf(a, b) gives local information of a signal f witha time-window

[b+ at∗ − a∆ψ, b+ at∗ + a∆ψ].


This window automatically narrows for small values of a (or highfrequencies) and widens for allowing a to be large (or small frequencies).

Now let us compare the local regularity of Fourier transforms andwavelets.

From the definition of the inverse Fourier transform (2.2) it is not hardto see, that if

|f(ω)| ≤ C

1 + |ω|n+1+ε,

then f ∈ Cn. Therefore, if f has a compact support, then f ∈ C∞. Thusthe smoothness of the signal f depends on decay of |f |.

In turn, one can see from (2.1), that the decay of |f | implies thesmoothness of f . For example, f = 1[−1,1] is discontinuous at t = ± 1,so |f(ω)| decays like |ω|−1 (not better). But in this case f(t) is smooth fort 6= ±1. Nevertheless this information can not be derived from the decay of|f(ω)|.

Now we consider a wavelet transform of the signal f :

Wψf(a, b) =1√|a|

∫R

f(t)ψ(t− b

a

)dt.

One of the main advantage of the wavelet transform is that it cancharacterize local regularities of the signal f : when the scale a goes to zero,the decay of the wavelet coefficients characterizes the regularity of the signalf in the neighborhood of b. One may say that the wavelet transform is asort of mathematical microscope analyzing arbitrary signals on variouslength scales around any time at time domain.

The following remarkable theorems were proved by M. Holschneider andPh. Tchamitchian [11]:

Theorem 3.1. Let the wavelet ψ satisfy the condition: (1 + |x|)ψ(x) ∈L1(R). If a bounded signal f is Holder continuous in x0 with exponent α ∈(0, 1], i.e.

|f(x0 + h)− f(x0)| ≤ C|h|α,then

|Wψf(a, x0 + b)| ≤ C|a|1/2(|a|α + |b|α).

Theorem 3.2. Let the wavelet ψ have a compact support. Suppose also thatf ∈ L2(R) is bounded and continuous. If for some β > 0 and α ∈ (0, 1),

|Wψf(a, b)| ≤ C|a|β+1/2 uniformly in b,


and|Wψf(a, x0 + b)| ≤ C|a|1/2

(|a|α +

|b|α

| log |b||),

then f is Holder continuous in x0 with exponent α.

Holschneider and Tchamitchian [11] investigated properties of theRiemann (1855) function

R(t) =∞∑n=1

1n2

sin(πn2t)

and presented a very simple proof, based on above Theorems, of results ofHardy and Gerver on differentiability of this function.

4 Convergence of wavelet transforms on theentire Lebesgue set of Lp-functions

Now we consider more general wavelet transforms, i.e. we take a differentfunction for the reconstruction than for the decomposition. More explicitly,let two functions ψj ∈ L2(R) satisfy the condition∫

|ω|−1|ψ1(ω)||ψ2(ω)|dω <∞, (4.1)

where ψj is the Fourier transform of ψj .The advantage of considering two different functions is that, functions

ψ1 and ψ2 may have very different properties. Moreover, both need not evenbe admissible: if ψ1 = O(ω) for ω → 0, then ψ2(0) 6= 0 is allowed.

The continuous wavelet transform of any signal f ∈ L2(R) by ψ1 isdefined by

Wψ1f(a, b) =∫ψa,b1 (t)f(t)dt,

whereψa,bj (t) = |a|−1/2ψj

(t− b

a

).

It is well known (see, for example, [12]) that if

Cψ1,ψ2 = 2π∫|ω|−1ψ1(ω)ψ2(ω)dω 6= 0,


thenS(A1, A2)f(t) =

C−1ψ1,ψ2

∫A1<|a|<A2

∞∫−∞

Wψ1f(a, b)ψa,b2 (t)da db

a2(4.2)

converges to f(t) when A1 → 0 and A2 →∞ in L2 norm.Obviously, from the convergence in L2 norm it does not follow the

pointwise convergence.We note that convergence of S(A1, A2)f(x) in those points, where a

given bounded L2-function f is continuous was investigated in [11], [12].The most general pointwise inversion formula was obtained in 1994 by

M. Rao, H. Sikic, R. Song [13] using the celebrated Carleson-Hunt theoremon convergence almost-everywhere of Fourier series. Namely, the authorsproved, if

ψj ∈ L2(R)⋂L1(R), ψj(0) = 0,

and f ∈ Lp(R), 1 < p <∞, then

limA1→0

S(A1,∞)f(t) = f(t) (4.3)

almost-everywhere.Recall that a point t ∈ R is a Lebesque point of a locally integrable

function f if

limr→0

12r

∫|t−x|<r

|f(x)− f(t)|dx = 0.

The Lebesgue theorem states that, given any locally integrable function f(hence for any f ∈ Lp(R), p ≥ 1) almost every t is a Lebesgue point. Theset of all Lebesgue points is called the Lebesgue set of f .

Theorem 4.1. ([1], [2]). Suppose that

1. ψ1, ψ2 ∈ L1(R) and ψ1(0) = ψ2(0) = 0;

2. tψj ∈ L1(R), j = 1, 2.

If f ∈ Lp(R), 1 ≤ p <∞, then the following equality

limA1→0, A2→∞

S(A1, A2)f(t) = f(t)

holds on the entire Lebesgue set of f .


Note, we suppose in Theorem 4.1, that the limits A1 → 0 and A2 →∞are independent (compare with (4.3)), i.e. we consider the doubly truncatedintegrals (4.2).

Condition (1) of Theorem 4.1 is the same as in [13] and adding thecondition xψ(x) ∈ L1(R) gives convergence on the entire Lebesgue set of f .In applications, in order to be well localized, wavelets have compact support,or they rapidly decay. Obviously these wavelets satisfy our extra conditionxψ(x) ∈ L1(R).

To prove the Theorem 4.1 we rewrite the partial sums in the form

S(A1, A2)f(x) = I(A1)f(x)− I(A2)f(x),

whereI(Aj)f(x) = A−1

j

∫ ∞

−∞K(x− y

Aj

)f(y)dy, j = 1, 2

and the Fourier transform of the function K(x) has the form

K(ξ) = (2π)12C−1

ψ1,ψ2

∫a>|ξ|

ψ1(a)ψ2(a)da

|a|.

We can prove the convergence of the partial sums if we have

K(·) ∈ L1(R). (4.4)

But using the condition xψj(x) ∈ L1(R) we can obtain even better result:

|K(x)| ≤ C

1 + x2.

Now let us consider the multidimensional continuous wavelet transforms(MCWT). We say ψ ∈ L2(Rn) is a wavelet if the admissibility condition

Cψ =∫Rn|ω|−n|ψ(ω)|2dω <∞

is satisfied. If ψ is a wavelet, we generate a doubly-indexed family ofwavelets:

ψa,b(x) = a−n/2ψ

(x− b

a

), (4.5)

where the parameter b ∈ Rn is a position parameter, whereas a > 0 is ascale parameter.


The MCWT of f ∈ L2(Rn) by ψ is defined by

Wψf(a, b) =∫Rn

ψa,b(x)f(x)dx. (4.6)

One may consider the following partially truncated integrals:

Sλf(x) = C−1ψ

∫ ∞

λ

∫Rn

Wψf(a, b)ψa,b(x)db da

an+1, (4.7)

or doubly truncated integrals:

S(A1, A2, B)f(x) =

C−1ψ

∫A1<|a|<A2

∫|b|≤B

Wψf(a, b)ψa,b(x)da db

an+1. (4.8)

It is well known (see, for example, [12]) that the doubly truncated integrals(consequently, the partially truncated integrals) converge in L2(Rn)-normto the given function f ∈ L2(Rn).

In the paper [14] M. Wilson considered the following partial sums of then-dimensional wavelet transforms:

S(Ej)f(t) = C−1ψ

∫ ∫Ej

Wψf(a, b)ψa,b(t)da db

an+1

and proved that if ψ satisfies some smoothness and decay conditions, thenS(Ej)f(t) converges to f in norm for any sequence of sets Ej ∈ Rn+1

which fills up Rn+1 and every f ∈ Lp(Rn), 1 < p < ∞ (note, in fact theauthor considered a different wavelet for the reconstruction than for thedecomposition).

In multidimensional case it is convenient to consider sphericallysymmetric wavelets: ψ is spherically symmetric wavelet if there is %(r) suchthat ψ(x) = %(|x|). Obviously ψ(ξ) is also spherically symmetric: for someη(t) one has

ψ(ξ) = η(|ξ|).

Therefore the admissibility condition will take the form

Cψ =∫ ∞

0

|η(t)|2

tdt <∞.


Let ψ ∈ L2(Rn) be a spherically symmetric wavelet. In the paper ofKangwei Li and Wenchang Sun [15] the authors proved the same result asin Theorem 1 for partial sums (4.8) by replacing our assumption xψj(x) ∈L1(R) with much easier one: let ψj(x) be radial functions and

|ψj(x)| ≤ ϕ(|x|), (4.9)

where ϕ is radial, positive, bounded, decreasing (as a function on (0,∞))function and ϕ(·) · ln(2 + | · |) ∈ L1(RN ). The function ϕ with theseproperties is called a radial log-majorant. The authors of the paper [15]in fact succeeded to prove (4.4) with this new assumption.

Sadahiro Saeki [16] investigated convergence of (4.7) to f nontangentiallyat every Lebesque point of f ∈ Lp(Rn), 1 < p < ∞. Note, since in thepaper of Kangwei Li and Wenchang Sun [15] the authors considered theconvergence of doubly truncated integrals, the assumptions on wavelets areslightly stronger than that in the paper of Sadahiro Saeki [16].

It should be noted, that the doubly truncated integrals S(A1, A2, B)f(x)are obviously different from the partially truncated one in (4.7); undercertain conditions, the partially truncated integrals (4.7) of any functionf is convergent to f in L1(Rn), which is impossible for (4.8) whenever∫Rn

f(x)dx 6= 0 (see, for example, [15]).If we compare the result of papers [15] and [16] (in one dimensional

case) with Theorem 4.1, we can see that there are wavelets which satisfythe conditions of Theorem 4.1 but not theorem of Kangwei Li and WenchangSun or Sadahiro Saeki.

In processing of n-dimensional signals there are two ways of waveletanalysis: to use one dimensional wavelets (so-called, Separable approach)or to use n-dimensional wavelets (Nonseparable approach). For example,spherically symmetric wavelets are one particular type of nonseparablewavelets.

In the papers [17] and [18] the authors investigated convergence ofseparable wavelet transforms in Pringsheim’s sense.

Let ψj(t), t ∈ R, one dimensional wavelets satisfy the conditions (3.1)and have a radial log-majorant (4.9). InRn consider wavelets ψ(x) = ψ1(x1)·ψ2(x2) · · · ψn(xn).


Wψf(a, b) =∫Rn

n∏j=1

ψaj ,bjj (xj)f(x)dx,


where a and b are vectors with coordinates aj and bj .We denote the partially truncated partial inverse transform by

S(A,B)f(x) = C−1ψ

∫A1<|a1|<B1

· · ·∫An<|an|<Bn

×

×∫Rn

Wψf(a, b)( n∏j=1

ψaj ,bjj (xj)a2j

)db da,

where A and B are vectors with corresponding coordinates.The authors proved convergence of S(A,B)f(x) to f ∈ Lp(Rn), 1 <

p < ∞ in Pringsheim’s sense (i.e. Aj → 0 and Bj → ∞ for all j) in theLp- norms and on the entire Lebesque set of a given function. It should benoted, that the authors studied the inverse wavelet transform S(A,B)f(x)with the help of the summability means of Fourier transforms, more exactly,they succeeded to represent the continuous wavelet transform S(A,B)f(x)in such a way, that summability of the Fourier integrals of f with thesuitable summation method became sufficient for convergence of its wavelettransform.

5 The Riesz means of spherically symmetricwavelet transforms

As we have seen above to have pointwise convergence of wavelet transformswavelets should have a sufficient decay conditions at infinity (for example,they should have a radial log-magorant (4.9)). In this section we willinvestigate the convergence of multiple continuous wavelet transformswithout any smoothness or decay conditions on wavelets ψ. To do this weintroduce the notion of the Riesz summation method for wavelet transforms.This method is widely used in classical harmonic analysis.

Let ψ be an arbitrary spherically symmetric wavelet with admissibilitycondition. For s ≥ 0 and λ > 0 we define the Riesz means of order s ofspherically symmetric wavelet transforms Sλf(x) (see (4.7)) as

Ssλf(x) = C−1ψ

∫a>λ

(1− λ2

a2

)sda

an+1

∫Rn

(Wψf)(a, b)ψa,b(x)db. (5.1)


Theorem 5.1. [6] For every f ∈ L2(Rn) and any s > 0

limλ→0+

Ssλf(x) = f(x)

holds almost everywhere on Rn.

To prove Theorem 5.1 we use the main idea of the paper [13]. In[13] the authors succeeded to represent the continuous wavelet transformS(A1,∞)f(x) in the following way

S(A1,∞)f(t) =∫R

ψ1(a)ψ2(a)|a|

da

∫|ξ|≤ |a|

A1

f(ξ)eitξdξ.

Thus to investigate convergence of wavelet transform of a given function fit is sufficient to investigate convergence of its Fourier integrals.

Let Esλf(x) be the Riesz means of the multiple Fourier integrals of f ∈L2(Rn):

Esλf(x) =∫|ξ|2<λ

(1− |ξ|2

λ

)sf(ξ)eixξdξ.

It is not hard to see that

Ssλf(x) =

∞∫0

Esa/λf(x)|η(a)|2 daa. (5.2)

Since any positive order Riesz means Esλf(x) converge almost-everywherefor f ∈ L2(Rn) (see, for example, [23]), then with the help of simplereasoning, from equation (5.2) we have our theorem.

With this connection we note that in the paper [24] the authorsintroduced a new wavelet-like transform (compare with (5.2)):

Tf(x; t) =

∞∫0

ρ(tη)Es1/tηf(x)g(η)dη,

where ρ(t) ≥ 0 is a weight function, g(t) is a wavelet function. Theysucceeded to investigate this transform only for s > n−1

2 and proved itsconvergence almost-everywhere to f ∈ Lp(Rn), 1 ≤ p <∞.

The following two theorems show that if a given function f is smoothenough then the Riesz means (5.1) of certain order converge uniformly tof . We first remind the definition of Liouville classes.


Definition 5.2. Let F denote the Fourier transform operator. Function fbelongs to Liouville class Llp(Rn), l > 0, p ≥ 1 , if

‖f‖Llp = ‖F−1(1 + |ξ|2)l/2Ff‖p <∞.

Remark. In addition to this definition, we assume to identify L0p with

classical Lp Banach spaces.Next we introduce the classical Riemann localization principle (well

known in harmonic analysis) for the wavelet transforms.

Definition 5.3. The localization principle for Ssλf(x) holds in Llp(Rn)

classes, if for any function f ∈ Llp(Rn) from the condition f(x) = 0, x ∈ Ωit follows lim

λ→0Ssλf(x) = 0 uniformly on any compact K ⊂ Ω.

Theorem 5.4. [4] (On localization.)Let f ∈ Llp(Rn) be a compactly supported function for p ≥ 1 and l ≥ 0.

If s ≥ 0 such that l + s ≥ n−12 , then lim

λ→0Ssλf(x) = 0, uniformly on any

compact K ⊂ Rn \ supp f .

Theorem 5.5. [4] (On uniform convergence.)Let f ∈ Llp(Rn) be a compactly supported function for p ≥ 1 and l > 0. If

s ≥ 0 such that l+ s ≥ n−12 and lp > n, then lim

λ→0Ssλf(x) = f(x) uniformly

on any compact K ⊂ Rn.

Proof of these theorems also based on equality (5.2). But note equality(5.2) holds true for any f ∈ L2(Rn). To prove Theorem 5.4 first we establishequality (5.2) for any compactly supported f ∈ Llp(R

n) and for every x ∈Rn \ suppf . Then we make use of similar results for the Fourier integrals(see, for example, [23]). Since in Theorem 5.5 we have lp > n, then to provethis theorem we first apply embedding theorems Llp → L

n22 and then make

use of equality (5.2).Now we introduce continuous wavelet transforms of distributions and

present some results on their convergence.As it is discussed in F. Jondral [20], within a self-contained signal theory,

distributions have to be taken into account, because without them notionslike impulse response or transmission function cannot be defined. There aresignal functions, for example f(t) = cosω(t), that are not of finite energy(and as a consequence they are not members of L2). Such signals are notrealizable in the physical sense but, from a theoretical point of view, they are


nevertheless of great importance. The use of the notion "impulse response"isalso inaccurate, since the Dirac impulse δ also is not found in L2.

Thus when we use wavelet analysis in solving initial value problems forpartial differential equations or in signal (image) processing we should beable to define wavelet transforms of distributions.

Wavelet transforms for distributions were considered for differentpurposes (see e.g. [21]) and [22]. In this paper we introduce continuouswavelet transforms of distributions with compact support and investigateconvergence properties of these transforms out of the distribution’s support.Of course, in general we cannot expect this convergence. Therefore weshould either consider wavelets which have rapidly vanishing derivativesor introduce some regularization methods. In the present work, like in theFourier analysis, we introduce the Riesz means of these transforms andfind sufficient condition for the order of Riesz means which guarantee theconvergence of the continuous wavelet transforms of distributions.

Let E(Rn) = C∞(Rn) be the space of infinitely differentiable functionson Rn with the topology of uniform convergence on compact sets offunctional sequences and all their derivatives. Let E′(Rn) denote the spaceof continuous linear functionals on E(Rn).

Recall, that this conjugate space consists of all compactly supportedSchwartz distributions D′(Rn) (i.e. the conjugate space to C∞0 (Rn)). Notethat any classical function with compact support can be considered as adistribution from E′(Rn). Moreover, considering f ∈ E′(Rn) as a tempereddistribution one may define the Fourier transform of f .

For any real l, by H l(Rn) we denote the Sobolev space. Recall, if f ∈E′(Rn), then we may define the norm of H l(Rn) as

‖f‖2Hl(Rn) ≡ ‖f‖2l,2 =∫Rn

(1 + |ξ|2)l|f(ξ)|2dξ,

where f is the Fourier transform of f ∈ E′(Rn). We note that for smoothfunctions we use the following usual definition

f(ξ) =∫Rn

f(t)eiξtdt.

In particular, if l > 0, then H l(Rn) is a closed subspace of L2(Rn) andH0(Rn) = L2(Rn).

It should be noted that each distribution f ∈ E′(Rn) has a "negativesmoothness i.e. it belongs to the Sobolev space H−l(Rn) with some l > 0.


For example, Dirac delta distribution δ(x), acting to functions ϕ ∈E(Rn) as < δ, ϕ >= ϕ(0), belongs to E′(Rn). It has the Fourier transformequal to 1, i.e. δ(ξ) = 1. Therefore

‖δ‖2l,2 =∫RN

(1 + |ξ|2)ldξ <∞,

for any l < −n/2. Hence δ ∈ H−l(Rn) for any l > n/2.To define continuous wavelet transforms of a distribution f ∈ E′(Rn)

one should consider only infinitely differentiable wavelets.

Definition 5.6. A function ψ(x) ∈ C∞(Rn)∩L1(Rn) is said to be a waveletif the following admissibility condition is satisfied∫

Rn|ω|−n|ψ(ω)|2dω <∞.

We shall consider here a class of spherically symmetric wavelets.Next we generate a doubly-indexed family of wavelets ψa,b(x) from a

spherically symmetric wavelet ψ by the formula (4.5).If we consider ψa,b(x) as a function of x, it belongs to C∞(Rn). Therefore

a distribution f ∈ E′(Rn) can act on this function.

Definition 5.7. The continuous wavelet transform of f ∈ E′(Rn)associated with ψ is defined by

(Tf)(a, b) =< f, ψa,b > .

Now we can introduce partial wavelet transforms Wλf(x).

Definition 5.8. For f ∈ E′(Rn) and λ > 0 we define partial wavelettransform Wλf(x) as

Wλf(x) = C−1ψ

∫ ∞

λ

da

an+1

∫Rn

(Tf)(a, b)ψa,b(x)db.

Finally for s ≥ 0 and λ > 0 we define the Riesz means of order s of thepartial wavelet transforms Wλf(x) as

W sλf(x) =

C−1ψ

∫a>λ

(1− λ2

a2

)sda

an+1

∫Rn

(Tf)(a, b)ψa,b(x)db. (5.3)


Theorem 5.9. [5] Let f ∈ E′(Rn)∩H−l(Rn) and l > 0. If s ≥ (n−1)/2+l,then

limλ→0+

W sλf(x) = 0

uniformly with respect to x ∈ K for any compact subset K ⊂ Rn \ supp f .

This, together with our results above, implies the following corollary.

Corollary 5.10. [5] Let f ∈ E′(Rn) ∩ H−l(Rn), l > 0 and let thedistribution f coincide with a continuous function g(x) in a domain Ω.If s ≥ (n− 1)/2 + l, then

limλ→0+

W sλf(x) = g(x)

uniformly on each compact set K ⊂ Ω.

To prove these statements we again make use of equality (5.2) and similarresults for the Fourier integrals, obtained by Sh. Alimov [24].

In classical harmonic analysis there is a well known notion calledgeneralized localization principle, which for the first time was formulatedby V.Il’in in [25]. For convenience, we give its definition for the Riesz meansW sλ .

Definition 5.11. We say that for the Riesz means W sλ of order s, the

generalized localization principle in function (or distribution) class F issatisfied, if for any function (or distribution) f ∈ F, the equality

limλ→∞

W sλf(x) = 0

is true for a.e. x ∈ Rn \ supp f .

This localization principle generalizes the classical Riemann localizationprinciple.

The following two theorems are new and their proof again is based onthe equality (5.2) and similar results, obtained for the Fourier integrals inthe papers [26] and [27].

Theorem 5.12. Let ψ be an arbitrary spherically symmetric wavelet. Forevery f ∈ Lp(Rn), 2 ≤ p < 2n

n−1 ,

limλ→0+

Sλf(x) = 0

holds almost everywhere on Rn \ supp f .


Theorem 5.13. Let ψ be a spherically symmetric wavelet defined as inDefinition 4 and f ∈ E′(Rn) ∩H−l(Rn), l > 0. If s ≥ l, then

limλ→0+

W sλf(x) = 0

holds almost everywhere on Rn \ supp f .

6 Almost everywhere convergence of wavelettransforms with square-symmetric spectrum

In this section we introduce a new class of multidimensional wavelets whichhas a very good convergence properties.

Define square norm by

‖x‖ = max1≤j≤n

|xi|.

Definition 6.1. A function ψ(x) ∈ L2(Rn) is said to be a wavelet withsquare-symmetric Fourier transform if

ψ(ξ) = ρ(‖ξ‖) (6.1)

and

Cψ =∫ ∞

0

|ρ(t)|2

tdt <∞. (6.2)

It should be noted, since ψ ∈ L2(Rn) then of course∫ ∞

0

|ρ(t)|2tn−1dt <∞.

If in R2 we define the function

Ψ(x1, x2) =1

π2x2

∫ ∞

0

ρ(t) cos tx1 sin tx2dt,

then it is not hard to verify that the class of all functions ψ ∈ L2(R2) withsquare-symmetric Fourier transform has the form ψ(x1, x2) = Ψ(x1, x2) +Ψ(x2, x1).


Let ψ ∈ L2(Rn) be a wavelet with square-symmetric Fourier transform.We generate a doubly-indexed family of wavelets ψa,b(x) from ψ by formula(4.5):

ψa,b(x) = a−n/2ψ

(x− b

a

),

where the parameter b ∈ Rn is a position parameter, whereas a > 0 is ascale parameter.


Wψf(a, b) =∫Rn

ψa,b(x)f(x)dx.

We denote the partial inverse transform by

Tλf(x) = C−1ψ

∫ ∞

λ

∫Rn

Wψf(a, b)ψa,b(x)db da

an+1. (6.3)

Since this is a new class of wavelets, then we should first make sure thatTλf(x) converges to a given function f ∈ L2(Rn) in L2 norm.

Theorem 6.2. [3] Let ψ ∈ L2(Rn) be a wavelet with square-symmetricFourier transform. Then for every f ∈ L2(Rn) one has

limλ→0+

||Tλf(x)− f(x)||2 = 0.

The proof of this theorem is based on a technique of I. Daubechies [12],which she used for spherically symmetric wavelets.

The next theorem states convergence almost everywhere of Tλf(x)without any smoothness or decay conditions at infinity on consideringwavelets.

Theorem 6.3. [3] Let ψ ∈ L2(Rn) ∩ L1(Rn) be a wavelet with square-symmetric Fourier transform. Then for every f ∈ Lp(Rn), 1 < p < ∞ onehas

limλ→0+

Tλf(x) = f(x)

almost everywhere on Rn.

To prove this theorem we first obtain a similar with (5.2) formula, whichconnects wavelet transforms Tλf(x) with square partial sums of the multipleFourier integrals:

Tλf(x) =


(2π)−nC−1ψ

∫ ∞

0

|ρ(a)|2

ada

∫||ω||<a/λ

f(ω)eiωxdω.

Then make use of the result on convergence almost everywhere of the squarepartial sums of the multiple Fourier integrals (see, for example, [23]).

References

1. R. R. Ashurov, Convergence of the continuous wavelet transforms onthe Lebesgue set of Lp functions, International Journal of Wavelets,Multiresolution and Information Processing (IJWMIP) 9(2011), no. 4, 676-683.

2. R.R. Ashurov, On the almost-everywhere convergence of the continuouswavelet transforms, Journal Proceedings of the Royal Society of Edinburgh,Proceedings of the Royal Society of Edinburgh, 142A, 1121-1129, 2012.

3. Almaz Butaev and R. R. Ashurov, On some class of nonseparable continuouswavelet transforms, Appl. Anal. 91 (12), 2257-2265, 2012.

4. R. R. Ashurov and Almaz Butaev, On spherically symmetric continuouswavelet transforms of functions from Liouville classes, International Journalof Mathematics and Computation, 11 (2011), no. J11, 111 - 117.

5. R. R. Ashurov and Almaz Butaev, On continuous wavelet transforms ofdistributions, Appl. Math. Letters, 24(2011), 1578-1583.

6. Almaz Butaev and R. R. Ashurov, On the almost-everywhere convergenceof the spherically symmetric continuous wavelet transforms, UzbekMathematical Journal, 4 (2014), 9-28.

7. Meyer, Y., Ondelettes: algorithmes et applications. Paris, Armand Colin,1992. (English translation: Wavelets: algorithms and applications. SIAMPress, Philadelphia, 1993).

8. J.P. Kahane and P.G. Limarie’-Rieusset, Fourier analysis and wavelets.Gordon and Breach Publishers, 1995.

9. Meyer, Y. (reviewer), An introduction to wavelets, by Charles K. Chui. Tenlectures on wavelets, by I. Daubechies. Bull. Amer. Math. Soc. 28 (1993),350-360.

10. C.K. Chui. An Introduction to Wavelets, volume 1 of Wavelet Analysis andits Applications. Academic Press, Boston, 1992.

11. M. Holschneider and Ph. Tchamitchian, Pointwise regularity of Riemanns"nowhere differentiable"function. Inventiones Mathematicae, 105(1991),157-175.

12. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.


13. M. Rao, H. Sikic, R. Song, Application of Carleson’s theorem to waveletinversion, Control and Cybernetics 23(1994), 761-771.

14. M. Wilson, How fast and in what sense(s) does the Calderon reproducingformula converge? J. Fourier Anal Appl. 16, 768-785 (2010).

15. Kangwei Li and Wenchang Sun, Pointwise Convergence of the CalderonReproducing Formula, J. Fourier Annal Appl, 18, 439-455, 2012.

16. Sadahiro Saeki, On the Reproducing Formula of Calderon, J. Fourier AnnalAppl, 2, 15-28, 1995.

17. F. Weisz, Inversion formulas for the continuous wavelet transform, ActaMath. Hungary, 138 (3), 2013, 237-258.

18. K. Szarvas and F. Weisz, Almost everywhere and norm convergence of theinverse continuous wavelet transform in Pringsheim’s sense, Acta Sci. Math.(Szeged), 82, 2016, 125-146.

19. I.A. Aliev and Estra Saglik, A new wavelet-like transform associated withthe Riesz-Bochner integral and relevant reproducing formula, Mediterr J.Math. 13 (2016), 4711-4722.

20. F. Jondral, Generalized functions in signal theory Acta ApplicandaeMathematicae, Vol. 63, 2000, 175Џ184.

21. M. Holschneider, Fractal wavelet dimensions and localization,Communications in Mathematical Physics 160 (1994) 457Џ473.

22. R. Pathak, The wavelet transform of distributions, The TohokuMathematical Journal 56 (2004) 411Џ421.

23. Sh.Alimov, R.Ashurov, A.Pulatov. Multiple Fourier Series and FourierIntegrals. Commutative Harmonic Analysis -IV. Springer-Verlag, pp 1-97(1992).

24. Sh.Alimov, On spectral decompositions of distributions, Dokl. Akad. Nauk,Vol 331, No 6, 661-662 (1993).

25. V.Ii’in, On a generalized interpretation of the principle of localization forFourier series with respect to fundamental systems of functions, Sib. Mat.Zh.,Vol 9, No 5, 1093-1106 (1968).

26. A.Carbery, F.Soria, Almost-everywhere convergence of Fourier integralsfor functions in Sobolev spaces, and L2 localization principle, RevistaMatematica Iberoamericana, Vol 4, No 2, 319-337 (1988).

27. R. R. Ashurov and Almaz Butaev, On generalized localization of Fourierinversion for distributions, Contemporary Mathematics, Vol. 672, 33-51(2016).


Ravshan Ashurov,Professor, Institute of Mathematics, Uzbek Academy of SciencesMirzo Ulugbek street, 81, 100170 Tashkent, Uzbekistan„ e-mail:[email protected]

Almaz Butaev,PhD student, Department of Mathematics and Statistics, ConcordiaUniversity, Montreal, Canada, e-mail: [email protected]

Certain relationships between q- product ... 27


Certain relationships between q- product identities,modular equations and combinatorial partition

identitiesM.P. Chaudhary, Getachew Abiye Salilew, Feyissa Kaba

Wakene

Abstract. The purpose of this paper to present two identities whichdepict interrelationships among q- product identities, modular equationsand combinatorial partition identities

Keywords: q- product identities, combinatorial partition identities,modular equation and Jacobi’s triple product identities.

Mathematics Subject Classification (2010): 5A30, 11F27, 11P83

1. Introduction and Preliminaries.

Throughout this paper, N, Z, and C denote the sets of positive integers,integers, and complex numbers, respectively, and N0 := N ∪ 0. Thefollowing q-notations are recalled (see, e.g., [4, Chapter 6]): The q-shiftedfactorial (a; q)n is defined by

(a; q)n :=

1 (n=0)∏n−1k=0(1− aqk) (n ∈ N),

(1.1)

where a, q ∈ C and it is assumed that a 6= q−m(m ∈ N0). We also write

(a; q)∞ :=∞∏k=0

(1−aqk)

=∞∏k=1

(1− aqk−1) (a, q ∈ C; |q| < 1). (1.2)

It is noted that, when a 6= 0 and |q| = 1, the infinite product in (1.2)diverges. So, whenever (a; q)∞ is involved in a given formula, the constraint|q| < 1 will be tacitly assumed.The following notations are also frequently used:

(a1, a2, · · · , am; q)n := (a1; q)n(a2; q)n · · · (am; q)n (1.3)

28 M.P. Chaudhary, Getachew Abiye Salilew, Feyissa Kaba Wakene

and(a1, a2, · · · , am; q)∞ := (a1; q)∞(a2; q)∞ · · · (am; q)∞. (1.4)

Ramanujan defined the general theta function f(a, b) as follows (see, fordetails, [3, p. 31, Eq.(18.1)] and [5]; see also [1]):

f(a, b) = 1 +∞∑n=1

(ab)n(n−1)

2 (an + bn)

=∞∑

n=−∞an(n+1)

2 bn(n−1)

2 = f(b, a) (|ab| < 1). (1.5)

We find from (1.5) that

f(a, b) = an(n+1)

2 bn(n−1)

2 f(a(ab)n, b(ab)−n) = f(b, a) (n ∈ Z). (1.6)

Ramanujan also rediscovered the Jacobi’s famous triple-product identity(see [3, p. 35, Entry 19]):

f(a, b) = (−a; ab)∞(−b; ab)∞(ab; ab)∞, (1.7)

which was first proved by Gauss.Several q-series identities emerging from Jacobi’s triple-product identity(1.7) are worthy of note here (see [3, pp. 36-37, Entry 22]):

φ(q) :=∞∑

n=−∞qn

2= 1+2

∞∑n=1

qn2

= (−q; q2)∞2(q2; q2)∞ =(−q; q2)∞(q2; q2)∞(q; q2)∞(−q2; q2)∞

; (1.8)

ψ(q) := f(q, q3) =∞∑n=0

qn(n+1)

2 =(q2; q2)∞(q; q2)∞

;

(1.9)

f(−q) := f(−q,−q2) =∞∑

n=−∞(−1)nq

n(3n−1)2

(1.10)

=∞∑n=0

(−1)nqn(3n−1)

2 +∞∑n=1

(−1)nqn(3n+1)

2 = (q; q)∞


Equation (1.10) is known as EulerвҐЄs Pentagonal Number Theorem. Thefollowing q-series identity:

(−q; q)∞ =1

(q; q2)∞=

1χ(−q)

(1.11)

provides the analytic equivalence of Euler’s famous theorem: The number ofpartitions of a positive integer n into distinct parts is equal to the numberof partitions of n into odd parts.We also recall the Rogers-Ramanujan continued fraction of R(q):

R(q) := q15H(q)G(q)

= q15f(−q,−q4)f(−q2,−q3)

= q15

(q; q5)∞(q4; q5)∞(q2; q5)∞(q3; q5)∞

=q

15

1+q

1+q2

1+q3

1+· · · (|q| < 1). (1.12)

Here G(q) and H(q) are widely investigated Roger-Ramanujan identitiesdefined by

G(q) :=∞∑n=0

qn2

(q; q)n=

f(−q5)f(−q,−q4)

=1

(q; q5)∞(q4; q5)∞=

(q2; q5)∞(q3; q5)∞(q5; q5)∞(q; q)∞

; (1.13)

H(q) :=∞∑n=0

qn(n+1)

(q; q)n=

f(−q5)f(−q2,−q3)

=1

(q2; q5)∞(q3; q5)∞

=(q; q5)∞(q4; q5)∞(q5; q5)∞

(q; q)∞; (1.14)

and the functions f(a, b) and f(−q) are given in (1.5) and (1.10),respectively. For a detailed historical account of (and for various investigateddevelopments stemming from) the Rogers-Ramanujan continued fraction(1.12) and identities (1.13) and (1.14), the interested reader may refer tothe monumental work [3, p. 77 et seq.] (see also [1, 4]).The following continued fraction was recalled in [6, p. 5, Eq. (2.8)] from anearlier work cited therein: For |q| < 1,

(q2; q2)∞(−q; q)∞ =(q2; q2)∞(q; q2)∞

(1.15)


=1

1−q

1+q(1− q)

1−q3

1+q2(1− q2)

1−q5

1+q3(1− q3)

1− · · ·;

(q; q5)∞(q4; q5)∞(q2; q5)∞(q3; q5)∞

=1

1+q

1+q2

1+q3

1+q4

1+q5

1+q6

1+· · · ; (1.16)

C(q) :=(q2; q5)∞(q3; q5)∞(q; q5)∞(q4; q5)∞

= 1 +q

1+q2

1+q3

1+q4

1+q5

1+q6

1+· · · . (1.17)

Andrews et al. [2] investigated new double summation hypergeometric q-series representations for several families of partitions and further exploredthe role of double series in combinatorial partition identities by introducingthe following general family:

R(s, t, l, u, v, w) :=∞∑n=0

qs(n2 )+tnr(l, u, v, w;n), (1.18)

where

r(l, u, v, w : n) :=[nu ]∑j=0

(−1)jquv(

j2)+(w−ul)j

(q; q)n−uj(quv; quv)j. (1.19)

The following interesting special cases of (1.18) are recalled (see [2, p.106,Theorem 3]; see also [1]):

R(2, 1, 1, 1, 2, 2) = (−q; q2)∞; (1.20)

R(2, 2, 1, 1, 2, 2) = (−q2; q2)∞; (1.21)

R(m,m, 1, 1, 1, 2) =(q2m; q2m)∞(qm; q2m)∞

. (1.22)

Here, in this paper, we aim to present certain interrelations betweenq-product identities, modular equations and combinatorial partitionidentities associated with the identities in (1.8) and (1.20)-(1.22).

2. Sets of Preliminary Results.

Here we recall the following results for the verification of the main resultsin sections (see [10,11]).

If X =φ(−q)φ(−q3)

and Y =φ(q)φ(q3)

,


then,X

Y+Y

X− 3XY

+XY = 0

(2.1)

If P =φ(−q)φ(−q3)

and Q =φ(−q4)φ(−q12)

,

then,[P 4

Q4+Q4

P 4

]−8[P 2− 3

P 2

]−[Q4+

9Q4

]+

+P 2

[Q4 +

3Q4

]− 1P 2

[Q4 +

27Q4

]+ 16 = 0 (2.2)

3. The Main Results.

Here we state and prove certain intersecting interrelationships betweenq- product identities, modular equations and combinatorial partitionidentities asserted by following theorem.

Theorem. Each of the following relationships holds true.

3R(3, 3, 1, 1, 1, 2)R(2, 2, 1, 1, 2, 2)2(−q3, q3, q6; q6)∞R(1, 1, 1, 1, 1, 2)(−q6,−q6,−q3; q6)∞(q; q)∞

=R(3, 3, 1, 1, 1, 2)(−q6,−q3,−q3; q6)∞(q; q)∞

R(1, 1, 1, 1, 1, 2)R(2, 1, 1, 1, 2, 2)2(−q6, q3, q6; q6)∞+

+R(1, 1, 1, 1, 1, 2)R(2, 1, 1, 1, 2, 2)2(−q6, q3, q6; q6)∞

R(3, 3, 1, 1, 1, 2)(−q6,−q3,−q3; q6)∞(q; q)∞+

+R(1, 1, 1, 1, 1, 2)(−q6,−q6,−q3; q6)∞(q; q)∞

R(3, 3, 1, 1, 1, 2)R(2, 2, 1, 1, 2, 2)2(−q3, q3, q6; q6)∞(3.1)

R(2, 2, 1, 1, 1, 2)R(4, 4, 1, 1, 1, 2)R(12, 12, 1, 1, 1, 2)2×

×

(q3; q6)2∞(q6; q12)∞(q12; q24)∞(q4, q4, q4; q8)∞(q; q2)2∞(q8; q8)∞(q12; q12)2∞(q12; q24)∞

2

+

+27

(q3, q3, q6; q6)∞(q12, q12, q24; q24)2∞(q, q, q2; q2)∞(q4, q4, q8; q8)2∞

2

= R(6, 6, 1, 1, 1, 2)R(8, 8, 1, 1, 1, 2)4×

×

(q; q2)2∞(q2; q4)∞(q4; q8)∞(q8; q16)∞(q12; q24)∞(q3; q6)2∞(q4; q8)2∞(q12; q12)∞(q16; q16)∞

4

+


+R(2, 2, 1, 1, 1, 2)R(12, 12, 1, 1, 1, 2)4×

×

(q3; q6)2∞(q4; q8)∞(q6; q12)∞(q12; q24)∞(q; q2)2∞(q4; q4)∞(q12; q12)∞

4

−

−8

(q, q, q2; q2)∞(q3, q3, q6; q6)∞

2

+ 24

(q3, q3, q6; q6)∞(q, q, q2; q2)∞

2

−

−

(q4, q4, q8; q8)∞(q12, q12, q24; q24)∞

4

− 9

(q12, q12, q24; q24)∞(q4, q4, q8; q8)∞

4

+

+

(q, q, q2; q2)∞(q4, q4, q8; q8)2∞(q3, q3, q6; q6)∞(q12, q12, q24; q24)2∞

2

+

+3R(4, 4, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2)R(12, 12, 1, 1, 1, 2)2×

×

(q; q2)2∞(q2; q4)∞(q4; q8)∞(q12, q12, q12; q24)∞(q3; q6)2∞(q24; q24)∞(q4; q4)2∞(q4; q8)∞

2

+ (4)2 (3.2)

Proofs: First of all we have to prove our identity (3.1). Applying identity(1.8) [for q = −q, q = −q3 and q = q3] and further using (1.20)-(1.21), aftersimplifications we get the values for x and y as:

X =φ(−q)φ(−q3)

=(−q6,−q3; q6)∞(q; q)∞

R(2, 1, 1, 1, 2, 2)R(2, 2, 1, 1, 2, 2)(q3; q3)∞(3.3)

Y =φ(q)φ(q3)

=R(1, 1, 1, 1, 1, 2)R(2, 1, 1, 1, 2, 2)(−q6, q3; q6)∞

R(2, 2, 1, 1, 2, 2)(−q3, q6; q6)∞(3.4)

using (3.3) and (3.4) into (2.1), arranging the terms and by little algebra,we get required result (3.1).Further, we have to prove our second identity (3.2). Applying identity (1.8)[for q = −q,−q3,−q4 and −q12], we get the values for P and Q, as:

P =φ(−q)φ(−q3)

=(q; q)∞(q; q2)∞

(q3; q3)∞(q3; q6)∞(3.5)

Q =φ(−q4)φ(−q12)

=(q4; q4)∞(q4; q8)∞

(q12; q12)∞(q12; q24)∞(3.6)

with the help of (3.5)-(3.6) and applying (1.20)-(1.22), we obtain thefollowing:»P 4

Q4+Q4

P 4

–= R(6, 6, 1, 1, 1, 2)R(8, 8, 1, 1, 1, 2)4×


×

(q; q2)2∞(q2; q4)∞(q4; q8)∞(q8; q16)∞(q12; q24)∞(q3; q6)2∞(q4; q8)2∞(q12; q12)∞(q16; q16)∞

ff4

+

+R(2, 2, 1, 1, 1, 2)R(12, 12, 1, 1, 1, 2)4×

×

(q3; q6)2∞(q4; q8)∞(q6; q12)∞(q12; q24)∞(q; q2)2∞(q4; q4)∞(q12; q12)∞

ff4

(3.7)»P 2 − 3

P 2

–=

(q, q, q2; q2)∞

(q3, q3, q6; q6)∞

ff2

− 3

(q3, q3, q6; q6)∞(q, q, q2; q2)∞

ff2

(3.8)»Q4 +

9

Q4

–=

(q4, q4, q8; q8)∞

(q12, q12, q24; q24)∞

ff4

+ 9

(q12, q12, q24; q24)∞

(q4, q4, q8; q8)∞

ff4

(3.9)

P 2

»Q4+

3

Q4

–=

(q, q, q2; q2)∞(q4, q4, q8; q8)2∞

(q3, q3, q6; q6)∞(q12, q12, q24; q24)2∞

ff2

+

+3R(4, 4, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2)R(12, 12, 1, 1, 1, 2)2×

×

(q; q2)2∞(q2; q4)∞(q4; q8)∞(q12, q12, q12; q24)∞(q3; q6)2∞(q24; q24)∞(q4; q4)2∞(q4; q8)∞

ff2

(3.10)

1

P 2

»Q4+

27

Q4

–= R(2, 2, 1, 1, 1, 2)R(4, 4, 1, 1, 1, 2)R(12, 12, 1, 1, 1, 2)2×

×

(q3; q6)2∞(q6; q12)∞(q12; q24)∞(q4, q4, q4; q8)∞(q; q2)2∞(q8; q8)∞(q12; q12)2∞(q12; q24)∞

ff2

+

+27

(q3, q3, q6; q6)∞(q12, q12, q24; q24)2∞

(q, q, q2; q2)∞(q4, q4, q8; q8)2∞

ff2

(3.11)

using (3.7)-(3.11) into (2.2), by arranging the terms and aftersimplifications, we get (3.2).

References

1. . M. Srivastava and M. P. Chaudhary, Some relationships betweenq -product identities, combinatorial partition identities and continued-fractionsidentities, Adv. Stud. Contemporary Math. (25)(3)(2015), 265-272.

2. . E. Andrews, K. Bringman and K. Mahlburg, Double series representationsfor Schur’s partition function and related identities, J. Combin. Theory Ser.A (132)(2015), 102-119.

3. . C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, Berlin,Heidelberg and New York, 1991.

4. . M. Srivastava and J. Choi, Zeta and q -Zeta Functions and AssociatedSeries and Integrals, Elsevier Science Publishers, Amsterdam, London andNew York, 2012.


5. . D. Baruah and J. Bora, Modular relations for the nonic analogues of theRogers Ramanujan functions with applications to partitions, J. Number Thy.(128)(2008), 175-206.

6. . P. Chaudhary, Generalization of Ramanujan’s identities in terms of q-products and continued fractions, Global J. Sci. Frontier Res. Math. DecisionSci. (12)(2)(2012), 53-60.

7. . P. Chaudhary, Generalization for character formulas in terms of continuedfraction identities, Malaya J. Mat. (1)(1)(2014), 24-34.

8. . P. Chaudhary, Some relationships between q -product identities,combinatorial partition identities and continued-fractions identities III,Pacic J. Appl.Math. (7)(2)(2015), 87-95.

9. . P. Chaudhary and J. Choi, Note on modular relations for Roger-Ramanujantype identities and representations for Jacobi identities, EastAsian Math. J.(31)(5)(2015), 659-665.

10. .S.M. Naika, K.S. Bairy and N.P. Suman, Recent advances in mathematics,RMS- Lecture note series no. 21,2015, 133-145.

11. . Bhargara, K.R. Vasuki and T.G. Sreeramamurthy, Some evaluationsofRamanujan’s cubic continued fraction, Indian J. Pure Apple. Math; 35(8)(2004) 1003-1025.

International Scientific Research and Welfare Organization NewDelhi 110018, India. e-mail:[email protected] of Mathematics, College of Natural andComputational Science Madda Walabu University, BaleRobe, Ethiopia.

Mean ergodic theorem for infinite measure 35

5


Mean ergodic theorem in function symmetric spacesfor infinite measureChilin V.I., Veksler A.S.

Abstract. Let E be a fully symmetric space of measurable functions on (0,∞),and let T : L1 + L∞ → L1 + L∞ be a Danford-Schwartz operator. It is proved

that the averages 1n+1

nPk=0

T k converge in E with respect to the strong operator

topology if and only if E is a separable space and L1 is not contained in E.Keywords: Symmetric function space, Dunford-Schwartz operator, mean

ergodic theoremMathematics Subject Classification (2010): 46E30, 37A30, 47A35

1 IntroductionThe well-known mean ergodic theorem asserts (see, for example, [5, ChapterVIII, §5]) that the Cesaro means An(T ) = 1

n

∑n−1k=0 T

k converge strongly ina reflexive Banach space (X, ‖·‖X) for every linear contraction T of X, thatis, given x ∈ X, there exists x ∈ X such that∥∥∥∥ 1

n

n−1∑k=0

T k(x)− x

∥∥∥∥X

→ 0 as n→∞.

Important examples illustrating the mean ergodic theorem are the Banachfunction spaces Lp, equipped with the standard norm ‖ · ‖p. If 1 < p <∞,then (Lp, ‖·‖p) is a reflexive space, and the averages An(T ) converge stronglyin Lp for any linear contraction T of Lp. For the spaces L1 and L∞, themean ergodic theorem is false, in general.

An important class of linear contractions in the spaces Lp are theDunford-Schwarz operators T (writing: T ∈ DS), that is such linearoperators T : L1 + L∞ → L1 + L∞ that

T (L1) ⊆ L1 and ‖T (f)‖1 ≤ ‖f‖1 for all f ∈ L1;

T (L∞) ⊆ L∞ and ‖T (f)‖∞ ≤ ‖f‖∞ for all f ∈ L∞.

36 Chilin V.I., Veksler A.S.

If T ∈ DS then T (E) ⊆ E and ‖T‖e→E ≤ 1 for every exact interpolationsymmetric space E for the Banach pair (L1, L∞) (see, for example, [7,Chapter II, §4, Sec.2]). Examples of such exact interpolation symmetricspace are the Orlicz, Lorentz, Marcinkiewicz, Musielak-Orlicz functionalspaces.

Recall that the class of exact interpolation symmetric spaces for theBanach pair (L1, L∞) coincides with the class of fully symmetric spaces [7,Chapter II, §4, Theorem 4.3].

Therefore, there is the problem of describing the class of fully symmetricspaces E for which the mean ergodic theorem with respect to the action ofan arbitrary T ∈ DS is valid.

It is known that, in the case of the non-atomic measure space ((0, a), ν),where 0 < a < ∞ and ν is the Lebesgue measure, the averages An(T )converge strongly in each separable symmetric space E(0, a) for everyT ∈ DS (see [9], [10]; also [11, Chapter 2, §2.1, Theorem 2.1.3]). At thesame time, for a non-separable fully symmetric space E = E(0, a) and

any function f ∈ E \ L∞(0, a)‖·‖E , there are T ∈ DS and a function f ,

equimeasurable with f , such that the averages An(T )(f) do not convergewith respect to the norm ‖ · ‖E [10].

Note that, for the separable symmetric space L1((0,∞), ν), there existsT ∈ DS such that the averages An(T ) do not converge strongly inL1((0,∞), ν).

In the present paper we consider the Dunford-Schwartz operators actingin a fully symmetric spaces E of real measurable functions defined on thehalf-line ((0,∞), ν). The following necessary and sufficient conditions for Eare established ensuring strong convergence in E of means An(T ) for alloperators T ∈ DS:

Theorem 1.1. Let (E, | · ‖E) be a fully symmetric space on ((0,∞), ν).The following conditions are equivalent:

(i). The averages 1n

∑n−1k=0 T

k converge strongly in E for each T ∈ DS;

(ii). (E, | · ‖E) is a separable space and L1 * E.

At the end of this paper we describe some (classes of) fully symmetricspaces for which the mean ergodic theorem holds/fails.


2 Preliminaries

Let ((0,∞), ν) be a σ-finite measure space with Lebesgue measure ν andlet L0 = L0(0,∞) be the algebra of (classes of) a.e. finite real-valuedmeasurable functions on ((0,∞), ν). Let L0(ν) be the subalgebra in L0

consisting of functions f ∈ L0 such that ν|f | > λ < ∞ for some λ > 0.We denote by Lp ⊆ L0(ν), 1 ≤ p ≤ ∞, the classical Banach space equippedwith the norm ‖ · ‖p.

If f ∈ L0(µ), then a non-increasing rearrangement of f is defined as

f∗(t) = infλ > 0 : ν|f | > λ 6 t, t > 0,

(see [7, Chapter II, §2]). Define in L0(ν) the Hardy-Littlewood-Polya partialorder, setting

f ≺≺ g ⇔s∫

0

f∗(t)dt 6

s∫0

g∗(t)dt for all s > 0.

A non-zero linear subspace E in L0(ν) with the Banach norm ‖ · ‖E iscalled symmetric (respectively, fully symmetric) space on ((0,∞), ν) if

f ∈ E, g ∈ L0(ν), g∗(t) ≤ f∗(t) for all t > 0 (respectively, g ≺≺ f),

implies that g ∈ E and ‖g‖E ≤ ‖f‖E .Every a fully symmetric space is a symmetric space. The converse,

generally speaking, is not true. At the same time, any separable symmetricspace on ((0,∞), ν) is a fully symmetric space.

Immediate examples of fully symmetric spaces are L1 ∩ L∞ with thenorm

‖f‖L1∩L∞ = max ‖f‖1, ‖f‖∞

and L1 + L∞ with the norm

‖f‖L1+L∞ = inf ‖g‖1 + ‖h‖∞ : f = g + h, g ∈ L1, h ∈ L∞

(see [7, Chapter II, §4]).For any symmetric space E are true the following continuous embeddings

[1, Chapter 2, §6, Theorem 6.6]

(L1 ∩ L∞, ‖ · ‖L1∩L∞) ⊂ (E, ‖ · ‖E) ⊂ (L1 + L∞, ‖ · ‖L1+L∞).


In addition, if E is a fully symmetric space and T ∈ DS, then Tf ≺≺ ffor each f ∈ E [7, Chapter II, S 3, Section 4]. Hence, T (E) ⊆ E and‖T‖E→E 6 1 [7, Chapter II, S 4, Section 2], i.e. E is an exact interpolationsymmetric spaces for the Banach pair (L1, L∞).

If (E, ‖ · ‖E) is a symmetric space, then the Kothe dual E× is defined as

E× = f ∈ L0(ν) : fg ∈ L1 for all g ∈ E,

and

‖f‖E× = sup |∫Ω

fg dν| : g ∈ E, ‖g‖E ≤ 1, where f ∈ E×.

It is known that (E×, ‖·‖E×) is a fully symmetric space [8, Part II, Chapter7, §7.2, Theorem 7.2.2]; in addition,

E ⊆ E××, (L1)× = L∞, (L∞)× = L1;

(L1 + L∞, ‖ · ‖L1+L∞)× = (L1 ∩ L∞, ‖ · ‖L1∩L∞);

(L1 ∩ L∞, ‖ · ‖L1∩L∞)× = (L1 + L∞, ‖ · ‖L1+L∞)

(see [8, Part II, Chapter 7]).Define

Rν = f ∈ L1 + L∞ : f∗(t) → 0 as t→∞.

It is clear that Rν = f ∈ L1 + L∞ : ν|f | > λ < ∞ for all λ > 0. By[7, Chapter II, §4, Lemma 4.4], (Rν , ‖ · ‖L1+L∞) is a fully symmetric space.

Let χE be the characteristic function of set E ⊆ (0,∞), and let 1 =χ(0,∞). The following Proposition gives necessary and sufficient conditionsfor the embedding of a symmetric space in Rν (see [2, Proposition 2.1]).

Proposition 2.1. A symmetric space E on ((0,∞), ν) is contained in Rν

if and only if 1 /∈ E. In particular, every separable symmetric space iscontained in Rν .

The sequence fn∞n=1 ⊂ L0 is called convergent to the function f ∈ L0

almost uniformly (a.u.) if for any ε > 0 there exists a measurable set A suchthat ν((0,∞) \A) ≤ ε and lim

n→∞‖(f − fn)χA‖∞ = 0.

It is clear that a.u.-convergence always implies a.e.-convergence andconvergence with respect to the measure ν.

In the proof of Theorem 3.1 we shall use the following version of theDunford-Schwartz individual ergodic theorem [3, Theorem 4.4].


Theorem 2.2. For any f ∈ Rν and T ∈ DS there exists a function f ∈ Rν

such that the sequence An(T )(f) a.u.-converges to f .

3 Mean ergodic theorem for fully symmetricspaces

We say that a fully symmetric space E satisfies the mean ergodic theorem(writing: E ∈ (MET)) if the averages An(T ) converge strongly in E forany operators T ∈ DS.

Theorem 3.1. If E is an not separable fully symmetric space on ((0,∞), ν),then E /∈ (MET).

Proof. By Theorem 4.8 [7, Chapter II, §4], there exists a positive numbera > 0, such that the symmetric space (E(0, a), ‖ · ‖E) on the interval((0, a), ν) is not separable space (here E(0, a) = χ(0,a) · E). Therefore, by

Theorem 2.5.1 [11], there are the function f0 ∈ (E(0, a) \L∞(0, a)‖·‖E ) and

the Dunford-Schwarz operator

T0 : L1((0, a), ν) + L∞((0, a), ν) → L1((0, a), ν) + L∞((0, a), ν)

such that the averages An(T0)(f0) do not converge with respect to the norm‖ · ‖E .

Define the Dunford-Schwartz operator

T : L1((0,∞), ν) + L∞((0,∞), ν) → T : L1((0,∞), ν) + L∞((0,∞), ν),

setting T (h) = T0(χ(0,a) · h), h ∈ L1((0,∞), ν) + L∞((0,∞), ν).If f = f0 · χ(0,a) + 0 · χ[a,∞), then f ∈ E(0,∞) and An(T )(f) =

An(T0)(f0). Consequently, the averages An(T )(f) do not converge in Ewith respect to the norm ‖ · ‖E .

We give one more sufficient condition such that E does not satisfy themean ergodic theorem.

Theorem 3.2. If E is a fully symmetric space on ((0,∞), ν) and E iscontained in L1, then E /∈ (SET).

Proof Consider the operator T ∈ DS defined by (Tf)(t) = f(t− 1) ift > 1, and (Tf)(t) = 0 if t ∈ [0, 1). Since

‖A2n(T )(χ(0,1))−An(T )(χ(0,1))‖1 =∥∥∥∥ 1

2nχ(0,2n) −

1nχ(0,n)

∥∥∥∥1

=


= n(1n− 1

2n) + n

12n

= 1,

it follows that the averages An(T )(χ(0,1))) do not converge with respect tothe norm ‖ · ‖1. Therefore L1 /∈ (СЭТ).

It is known [8, Chapter 6, §6.1, Proposition 6.1.1] that the embeddingE1 ⊂ E2 of two symmetric spaces (E1, ‖·‖E1) and (E2, ‖·‖E2) is continuous;i.e. there is a constant c > 0 such that ‖f‖E2 6 c‖f‖E1 for all f ∈ E1.

Since E ⊆ L1, it follows that ‖f‖1 6 c‖f‖E for all f ∈ E andsome constant c > 0. Consequently, the sequence An(T )(χ(0,1)) can notconverges in the space (E, ‖ · ‖E), which implies the absence of strongconvergence of averages An(T ) in (E, ‖ · ‖E). Thus, E /∈ (СЭТ).

The fundamental function of symmetric space E is defined by theequality ϕE(t) = ‖χ(0,t]‖E . It is known that the function ϕE(t) isquasi-concave (see [7, Chapter II, §4, Theorem 4.7]), in particular, ϕE(t)increases, and the function ϕE(t)

t decreases [7, Chapter II, §1, Definition1.1]. Consequently, there are limits

α(E) = limt→+∞

ϕE(t)t

and β(E) = limt→+0

ϕE(t) = ϕE(+0).

We note that

α(L1) = 1, β(L∞) = 1, α(Lp) = 0, 1 < p 6 ∞, β(Lp) = 0, 1 6 p <∞.

We need the following necessary and sufficient conditions for the embeddingof a symmetric space E in the spaces L∞ and L1.

Proposition 3.3. (i). E ⊆ L∞ if and only if β(E) > 0;(ii). E ⊆ L1 if and only if α(E) > 0;(iii). E ⊆ L1 if and only if L∞ ⊆ E×, where E× is the Kothe dual

space.

Proof (i). If E ⊆ L∞, then there is a positive number c0 > 0 such that‖f‖∞ 6 c0‖f‖E for all f ∈ E [8, Chapter 6, § 6.1, Proposition 6.1.1].Consequently,

ϕE(t) = ‖χ(0,t]‖E >‖χ(0,t]‖∞

c0=

1c0,

and β(E) = limt→+0

ϕE(t) > 1c0> 0.

If E * L∞, then there is a positive unbounded function f ∈ (E\L∞),in particular, ν(An) > 0, where An =

f > n

, n ∈ N. Choose a sequence


Bn ⊆ An such that Bn ⊇ Bn+1, 0 < ν(Bn) < ∞ and limn→∞

ν(Bn) = 0.Then

n · β(E) 6 n‖χBn‖E = ‖n · χBn‖E 6 ‖f‖E <∞ ∀ n ∈ N.

Thus β(E) = 0.(ii). If E ⊆ L1, then there is a positive number c1 > 0 such that ‖f‖1 6

c1‖f‖E for all f ∈ E [8, Chapter 6, §6.1, Proposition 6.1.1]. Consequently,

ϕE(t)t

=‖χ(0,t]‖E

t>‖χ(0,t]‖1c1 · t

=1c1,

andα(E) = lim

t→+∞

ϕE(t)t

>1c1> 0.

Suppose now that α(E) > 0. By [7, Chapter II, §4, Inequality (4.6)] we havethat

‖f∗ · χ(0,t]‖1 6t

ϕE(t)· ‖f∗ · χ(0,t]‖E

for all f ∈ E. Since ϕE(t)t > α(E) > 0, t > 0, it follows that t

ϕE(t) 6 1α(E) .

Consequently,

‖f∗ · χ(0,t]‖1 6‖f∗ · χ(0,t]‖E

α(E)6‖f∗‖Eα(E)

,

and ‖f‖1 6 ‖f‖Eα(E) <∞ for all f ∈ E, that is E ⊆ L1.

(iii). If E ⊆ L1, then+∞∫0

|1 · f |dν = ‖f‖1 < ∞ for all f ∈ E. Thus

1 ∈ E× and L∞ ⊆ E×.If L∞ ⊆ E×, then 1 ∈ E×, that is the linear functional

ϕ(f) =

+∞∫0

1 · f dν, f ∈ E,

is bounded on E. Consequently,

‖f‖1 =

+∞∫0

1 · |f |dµ = ϕ(|f |) 6 ‖ϕ‖‖f‖E <∞

for every function f ∈ E, that is E ⊆ L1.


Corollary 3.4. E * L1 ⇔ L∞ * E× ⇔ 1 /∈ E×.

Let f ∈ Rν and T ∈ DS. By Theorem 2.2 there exists a function f ∈ Rν

such that the sequence An(T )(f) a.u.-converges to f . Define the mappingP : Rν → Rν , setting

P (f) = f = (a.u.)− limAn(T )(f), f ∈ Rν .

It is clear that P is a linear map. Since the closed balls in (L1, ‖·‖1) is closedwith respect to measure convergence [6, Chapter IV, §3] and ‖An(T )(f)‖1 6‖f‖1 for all f ∈ L1, it follows that ‖P (f)‖1 6 ‖f‖1 for every function f ∈ L1.Consequently, ‖P‖L1→L1 6 1.

Similarly, if f ∈ L1 ∩ L∞, then ‖An(T )(f)‖∞ 6 ‖f‖∞. Therefore,the a.u.-convergence An(T )(f) → P (f) implies that ‖P (f)‖∞ 6 ‖f‖∞.According to [2, Theorem 3.1], there exists a unique operator P ∈ DS

such that P (f) = P (f) for all f ∈ Rν , in particular, ‖P‖Rν→Rν 6 1. Inaddition, by the classical mean ergodic theorem for the space L2, we havethat ‖An(T )(f)− P (f)‖L2 → 0 as n→∞ for any function f ∈ L2.

The following Proposition is a variant of the mean ergodic theorem forfully symmetric space (Rν , ‖ · ‖L1+L∞).

Proposition 3.5. If T ∈ DS, then ‖An(T )(f)−P (f)‖L1+L∞ → 0 for allf ∈ Rν .

Proof. Since supn>1

‖An(T )‖L1+L∞→L1+L∞ 6 1, ‖P‖L1+L∞→L1+L∞ 6 1,

L1 ∩ L∞ is dense in the Banach space (Rµ, ‖ · ‖L1+L∞) and ‖An(T )(f)−P (f)‖L2 → 0 (and hence ‖An(T )(f)− P (f)‖L1+L∞ → 0)) for any functionf ∈ L1 ∩ L∞ ⊂ L2, it follows that ‖An(T )(f) − P (f)‖L1+L∞ → 0 for allf ∈ Rµ.

Corollary 3.6. P 2 = P and (TP )(f) = P (f) = (PT )(f) for all f ∈ Rν .

Proof Since

An(T )− n

n+ 1An−1(T ) =

1n+ 1

n∑k=0

T k − n

n+ 1(1n

n−1∑k=0

T k) =Tn

n+ 1

and

(I − T )An−1(T ) = (I − T )1n

n−1∑k=0

T k =I − Tn

n,


it follows that

(I − T )An−1(T )(f) =I − Tn

n(f) =

I

n− (An(T )(f)− n

n+ 1An−1(T )(f))

a.u.−→ 0.

On the other hand,

TAn−1(T )(f) = T (1

n

n−1Xk=0

T k(f)) =1

n

nXk=1

T k(f) =1

n

n−1Xk=0

T k(Tf)п.р.−→ P (T (f)).

Consequently,

(I − T )An−1(T )(f) = An−1(T )(f)− TAn−1(T )(f)п.р.−→ P (f)− P (T (f)).

This means that (PT )(f) = P (f) for all f ∈ Rν .By Proposition 3.5 we have that ‖An(T )(f)−P (f)‖L1+L∞ → 0 for each

function f ∈ Rν . Consequently,

(TP )(f) = T (‖ · ‖L1+L∞ − limn→∞

(An(T )(f))) = ‖ · ‖L1+L∞ − limn→∞

T (An(T )(f)) =

‖ · ‖L1+L∞ − limn→∞

1n+ 1

n+1∑k=1

T k(f) =

= ‖ · ‖L1+L∞ − limn→∞

(n+ 2n+ 1

1n+ 2

n+1∑k=0

T k(f)− 1n+ 1

f) = P (f).

Therefore (TP )(f) = P (f) = (PT )(f) for all f ∈ Rν . Thus, An(T )·P = P ,and P 2 = P .

We need the following property of symmetric spaces [4, Proposition 2.2].

Proposition 3.7. Let (E, ‖ · ‖E) be a separable symmetric space and letE× ⊆ Rν . If fn, g ∈ E, fn ≺≺ g, n ∈ N, and fn → 0 with respect to themeasure ν, then ‖fn‖E → 0 as n→∞.

Proof Now we give the proof of Theorem 3.1. The implication (i) ⇒(ii) in Theorem 3.1 follows from Theorems 3.2 and 4.3.

(ii) ⇒ (i). If P (f) = f ∈ E, then Corollary 2 implies that

An(T )(f) = An(T )(P (f)) =1

n+ 1

n∑k=0

T k(P (f)) = P (f) = f. (3.1)

Let f ∈ E and g = f − P (f). Then P (g) = P (f) − P 2(f) = 0 (see 3.6).Since (E, ‖ · ‖E) is a separable symmetric space, it follows that E ⊂ Rµ


(see Proposition 2.1). Consequently, An(T )(g) → 0 with respect to themeasure ν (see Theorem 2.2). In addition, An(T )(g) ≺≺ g, n ∈ N. ByCorollary 3.4, we have 1 /∈ E×, that is E× ⊆ Rν (see Proposition 2.1).Thus ‖An(T )(g)‖E → 0 (see Proposition 3.7).

Now using Corollary 3.6, the equality (3.1) and the equality An(T )(g) =An(T )(f)−An(T )(P (f)), we get that ‖An(T )(f)− P (f)‖E → 0.

The Proposition 3.3 and Theorem 3.1 imply the following corollary.

Corollary 3.8. If E is a fully symmetric space on ((0,∞), ν), then E ∈(СЭТ) if and only if E is a separable space and α(E) = 0.

It is known that a fully symmetric space (E, ‖ · ‖E) is a separable spaceif and only if L1 ∩ L∞ is dense in (E, ‖ · ‖E) and ϕE(+0) = 0 [7, ChapterII, §4, Theorem 4.8]. Consequently, Theorem 3.1 implies the following

Corollary 3.9. If E is a fully symmetric space on ((0,∞), ν), then E ∈(СЭТ) if and only if lim

t→+∞ϕE(t)t = 0, ϕE(+0) = 0 and L1 ∩ L∞ is dense

in (E, ‖ · ‖E).

4 Applications.In this section we give examples of fully symmetric spaces E for which theinclusion E ∈ (SET) is true.

1. Let Φ be the Orlicz function, that is Φ : [0,∞) → [0,∞) is a convex,continuous at zero and nondecreasing function, for which Φ(0) = 0 andΦ(t) > 0, t > 0. Consider the Orlicz space

LΦ =

f ∈ L0(ν) :

∞∫0

Φ(|f |λ

)dν <∞ for some λ > 0

.

Let ‖f‖Φ = infλ > 0 :∞∫0

Φ(|f |λ

)dν 6 1 be the Luxembourg norm on

the Orlicz space LΦ.It is known that an Orlicz space (LΦ, ‖ · ‖Φ) is a fully symmetric space,

in addition, LΦ is a separable space if and only if β(LΦ) = limt→+0

ϕLΦ(t) = 0

[8, Part IV, § 14.3, Corollary 14.2.4].According to [8, Part IV, §16, Theorem 16.2.1], the inclusion LΦ ⊆ L1

holds if and only if Φ(t) 6 a · t for some a > 0 and all t > 0, which is


equivalent to the validity of inequality limt→∞

Φ(t)t < ∞. Thus, Theorem 3.1

implies the following criterion for the validity of mean ergodic theorem inOrlicz spaces.

Theorem 4.1. (LΦ, ‖ · ‖Φ) ∈ (MET) if and only if β(LΦ) = 0 andlimt→∞

Φ(t)t = ∞.

2. Let ψ be an increasing and concave non-zero function on[0,∞), ψ(0) = 0, and let

Λψ =f ∈ L0(ν) : ‖f‖Λψ =

∫ ∞

0

f∗(t) dψ(t) <∞,

be a Lorentz space. It is known that a Lorentz space (Λψ, ‖ · ‖Λψ ) is a fullysymmetric space, in addition, (Λψ, ‖ · ‖Λψ ) is a separable space if and onlyif ψ(+0) = 0 and ψ(+∞) = +∞ [7, Chapter II, §5, Lemma 5.1]. BesidesϕΛψ (t) = ψ(t). Thus, Theorem 3.1 implies the following criterion for thevalidity of mean ergodic theorem in Lorentz spaces.

Theorem 4.2. (Λψ, ‖·‖Λψ ) ∈ (MET) if and only if ψ(+0) = 0, ψ(+∞) =+∞ and lim

t→+∞ψ(t)t = 0.

References

1. C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press Inc.(1988).

2. V. Chilin, D. Comez and S. Litvinov. Pointwise ergodic theorems insymmetric spaces of measurable functions. ArXiv:1612.05802v1 [math.FA],17 Dec. 2016. 16 pp.

3. V. Chilin and S. Litvinov. Noncommutative individual ergodic theorems .ArXiv:1607.03452v4. [math.OA]. 22 May 2017. 18 pp.

4. P. G. Dodds, T. K. Dodds, F. A. Sukochev. Banach-Saks properties insymmetric spaces of measurable operators. Studia Math. V. 178, 2007, 125–166.

5. N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory,John Willey and Sons, 1988.

6. L.V. Kantorovich, G.P. Akilov, Functional Analysis, M.: "Nauka 1977.

7. S.G. Krein, Ju.I. Petunin, and E.M. Semenov, Interpolation of LinearOperators, Translations of Mathematical Monographs, Amer. Math. Soc.,V. 54. 1982.


8. B.A. Rubshtein, G.Ya. Grabarnik, M.A. Muratov, Yu.S. Pashkova,Foundations of Symmetric Spaces of Measurable Functions. Lorentz,Marcinkiewicz and Orlicz Spaces. Springer International PublishingSwitzerland. 2016.

9. A. S. Veksler, Ergodic theorem in symmetric spaces, Siberian Math. J., V.26(4). 1985, 189–191.

10. A.S. Veksler, A.L. Fedorov, Statistic ergodic theorem in non-separablesymmetric function spaces, Siberian Math. J., V. 29(3). 1988, 183–185.

11. A.S. Veksler, A.L. Fedorov, Symmetric Spaces and Statistic ErgodicTheorems for Automorphisms and Flows, "FAN"Publishing, UzbekistanAcademy of Sciences, 2016.

Chilin VladimirUzbek National University, Tashkent, 100174, Uzbekistan, e-mail: [email protected] AleksandrInstitute of Mathematics Uzbekistan Academy of Sciences,Tashkent, Usbekistan, e-mail: [email protected]

2-Local derivations on real von Neumann algebras 47


2-Local derivations on real von Neumann algebrasDadakhodjaev R.A.

Abstract. The paper is devoted to the description of 2-local derivations onreal von Neumann algebras. It is proved that on arbitrary real von Neumannalgebra (except the case of type III) each 2-local derivation is a derivation.Analogously to the complex case, at first the result is proved for B(H) (where His a Hilbert space with dim(H) ≤ ∞), and then for semi-finite real von Neumannalgebras.

Keywords: Real von Neumann algebra, Derivation, 2-local derivationMathematics Subject Classification (2010): 46L36, 46L37, 46L57,

46C15

1 Introduction.Given an algebra A, a linear operator D : A → A is called a derivation, ifD(xy) = D(x)y + xD(y), for all x, y ∈ A. Each element a ∈ A implementsa derivation Da on A defined as Da(x) = [a, x] = ax − xa, x ∈ A. Suchderivations are said to be inner derivations. A map ∆ : A → A (not linearin general) is called a 2-local derivation, if for every x, y ∈ A, there exists aderivation Dx,y : A → A such that ∆(x) = Dx,y(x) and ∆(y) = Dx,y(y).

In the paper [1] P.Semrl introduced the notion of 2-local derivations anddescribed 2-local derivations on the algebra B(H) of all bounded linearoperators on the infinite-dimensional separable (complex) Hilbert spaceH. A similar description for the finite-dimensional case appeared later in[2]. In the papers [4], [5] and [6] the authors extended the Semrl’s resultfor arbitrary finite, semi-finite and purely infinite von Neumann algebras,respectively. In the present paper step by step we will prove analogous resultfor real B(H) and then for finite, semi-finite real von Neumann algebras.

2 Preliminaries.Let B(H) be the algebra of all bounded linear operators on a complexHilbert space H. A weakly closed *-subalgebra M containing the identityoperator 1I in B(H) is called a W*-algebra. A real *-subalgebra R ⊂ B(H)is called a real W*-algebra if it is closed in the weak operator topology,

48 Dadakhodjaev R.A.

1I ∈ R and R ∩ iR = 0. A real W*-algebra R is called a real factor ifits center Z(R) consists of the elements λ1I, λ ∈ R. We say that a realW*-algebra R is of the type Ifin, I∞, II1, II∞, or IIIλ, (0 ≤ λ ≤ 1)if the enveloping W*-algebra R + iR has the corresponding type inthe ordinary classification of W*-algebras. A linear mapping α of analgebra into itself with α(x∗) = α(x)∗ is called an *-automorphismif α(xy) = α(x)α(y); it is called an involutive *-antiautomorphism ifα(xy) = α(y)α(x) and α2(x) = x. If α is an involutive *-antiautomorphismof a W*-algebra M , we denote by (M,α) the real W*-algebra generated byα, i.e. (M,α) = x ∈ M : α(x) = x∗. Conversely, every real W*-algebraR is of the form (M,α), where M is the complex envelope of R and α is aninvolutive *-antiautomorphism of M (see [7, 8]). Therefore we shall identifyfrom now on the real von Neumann algebra R with the pair (M,α).

A trace on a (complex or real) W*-algebra N is a linear function τ onthe set N+ of positive elements of N with values in [0,+∞], satisfyingτ(uxu∗) = τ(x), for an arbitrary unitary u and for any x in N .

The trace τ is said to be faithful if τ(x) > 0 for all non-zero x ∈ N+;finite, if τ(1I) < +∞; semifinite, if given any x ∈ N+ there is a nonzeroy ∈ N+, y ≤ x with τ(y) < +∞. The trace τ is normal if τ(xγ) τ(x) forevery net xγ monotone increasing to x, (xγ , x ∈ N+).

It is easy to shown that every derivation of real W*-algebra is an inner.Indeed, let D : R→ R be a derivation. D can be extended by the linearityto a derivation on M = R+ iR as D(x+ iy) = D(x) + iD(y). Since D is aninner there is an element z = a+ib (a, b ∈ R) such thatD(x+iy) = [z, x+iy]for all x, y ∈ R. Hence D(x) = D(x) = [z, x] = [a + ib, x] = [a, x] + i[b, x].Therefore from D(x) ∈ R we have b = 0, i.e. z = a ∈ R. Thus D(x) = [a, x].

3 2-Local derivations on real B(H).

Let H be a real Hilbert space, and let B(H) be the algebra of all linearbounded operators on H. Let us at first to consider a finite-dimensionalcase, i.e. let dim(H) = n < ∞. Then as it is known that B(H) = Mn(R).Note that, the idea for the proofs of main theorems of paragraph comesfrom that of [1].

Theorem 3.1. Every 2-local derivation ∆ : Mn(R) → Mn(R) is aderivation.


Proof. Let e1, e2, . . . , en be the standard basis for Rn. We define twomatrices A,N ∈Mn = Mn(R) by

A =

12 0 . . . 00 1

22 . . . 0...

......

...0 . . . 0 1

2n

, N =

0 1 0 . . . 00 0 1 . . . 0...

......

......

0 0 . . . 1 00 0 . . . 0 10 0 . . . 0 0

A straightforward calculation shows that C ∈Mn commutes with A if andonly if it is diagonal, and if U commutes with N , then U is of the formUek = λke1 + λk−1e2 + · · ·+ λ1ek (k = 1, 2, . . . , n), for some λknk=1 ⊂ R.That is, U is of the form

U =

λ1 λ2 λ3 . . . λn0 λ1 λ2 . . . λn−1

0 0 λ1 . . . λn−2

......

......

.... . . . . . . . . λ1 λ2

0 0 . . . 0 λ1

Since every derivation of Mn is an inner there exists an element R ∈ Mn

such that ∆(A) = [R,A] and ∆(N) = [R,N ]. Replacing ∆ by the mapping∆ − ∆A,N if necessary, we can assume with no loss of generality that∆(A) = ∆(N) = 0, where ∆A,N (·) = [R, ·]. In order to prove the theoremenough to show, that ∆ ≡ 0.

Let T ∈ Mn be an arbitrary matrix. Then for A and T there existsS ∈Mn such that ∆(A) = [S,A] = 0 and ∆(T ) = [S, T ]. Since S commuteswith A it is diagonal. Analogously, for N and T there exists U ∈ Mn suchthat ∆(N) = [U,N ] = 0 and ∆(T ) = [U, T ]. Since U commutes with Nit has the above form. Thus for every T ∈ Mn there exist diagonal S andmatrix U of the above form, depending on T , such that

∆(T ) = TS − ST = TU − UT. (∗)

Let Ei,jni,j=1 be the system of matrix units of Mn. Then for any fixed iand j, according to (∗) (i.e. we apply (∗) to everyone Ei,j), we have

∆(Ei,j) = Ei,jS − SEi,j = Ei,jU − UEi,j ,


for some S = diag(ν1, . . . , νn) and U of the above form. Since

Ei,jS − SEi,j = (νj − νi)Ei,j(j=i)= 0, Ei,jU − UEi,j

(j 6=i)= 0,

it follows that ∆(Ei,j) = 0. Let R ∈Mn such that

∆(Ei,j) = ∆Ei,j ,T (Ei,j) = [R,Ei,j ] and ∆(T ) = ∆Ei,j ,T (T ) = [R, T ].

Using [R,Ei,j ] = 0 we get Ei,j(RT − TR)Ei,j = REi,jTEi,j − Ei,jTEi,jR,i.e. Ei,j∆(T )Ei,j = ∆Ei,j ,T (Ei,jTEi,j). Noting that Ei,j is the rank oneoperator ei ⊗ ej , we then have

Ei,j∆(T )Ei,j = ∆Ei,j ,T (Ei,jTEi,j) = 〈Tei, ej〉∆Ei,j ,T (Ei,j) = 〈Tei, ej〉∆(Ei,j) = 0.

From this equation it follows that 〈∆(T )ei, ej〉Ei,j = 0, and hence∆(T ) = 0.

Now, let us to consider an infinite-dimensional case, i.e. let H be aninfinite-dimensional separable real Hilbert space, and let B(H) be thealgebra of all linear bounded operators on H. Then the following theoremis carried out.

Theorem 3.2. Every 2-local derivation ∆ : B(H) → B(H) is a derivation.

Proof. Below we will generalize the proof of the previous theorem. Let en :n = 1, 2, . . . be an orthonormal basis in H. For any x, y ∈ H we denote theinner product of these two vectors by y∗x, while xy∗ denotes the rank oneoperator given by (xy∗)z = (y∗z)x. We define two operators A,N ∈ B(H)by

A =∞∑n=1

12nene

∗n, N(e1) = 0, N(en) = en−1, n = 2, 3, . . . .

For any bounded sequence (λn) ⊂ R we define D(Λ) ∈ B(H) byD(Λ)(en) = λnen. It is easy to see T ∈ B(H) commutes with A ifand only if it is of the form T = D(Λ) for some bounded sequence(λn) ⊂ R. Analogously, T commutes with N if and only if there exists asequence Γ = (ηn) ⊂ R, such that T = U(Γ), where U(Γ) is defined byU(Γ)(en) = ηne1 + ηn−1e2 + . . .+ η1en.

Now, we will consider A and N . Since every derivation of R is an innerthere is an element S ∈ R such that ∆(A) = [S,A] and ∆(N) = [S,N ].Replacing ∆ by the mapping ∆ − ∆A,N if necessary, we can assume with


no loss of generality that ∆(A) = ∆(N) = 0, where ∆A,N (T ) = [S, T ]. Inorder to prove the theorem enough to show, that ∆ ≡ 0.

Let Pn : H → spane1, e2, . . . en be the Hermitian projection onthe linear span of the set e1, e2, . . . en, n ∈ N. Let T ∈ B(H) withPnTPn = T and rank(T ) = n. Since ∆ is a 2-local derivation for elementsA and T there is an element R ∈ B(H) such that ∆(A) = [R,A] and∆(T ) = [R, T ]. But ∆(A) = 0 and so R commutes with A. Consequently,we have R = D(Λ) for some complex sequence Λ, which further yields

Pn∆(T )Pn = Pn[D(Λ), T ]Pn = Pn(D(Λ)T − TD(Λ))Pn =

D(Λ)PnTPn−PnTPnD(Λ) = D(Λ)T−TD(Λ) = [D(Λ), T ] = ∆(T ).

If we consider operators N and T we get an element F ∈ B(H) suchthat ∆(N) = [F,N ] and ∆(T ) = [F, T ]. Since ∆(N) = 0 an element Fcommutes with N , and therefore exists a sequence Γ with F = U(Γ). Thenusing Pn∆(T )Pn = ∆(T ), we obtain

0 = Pn∆(T )(I − Pn) = Pn(U(Γ)T − TU(Γ))(I − Pn) =

= Pn(U(Γ)PnTPn − PnTPnU(Γ))(I − Pn),

But obviously, rank(PnU(Γ)PnT ) = n, and so, we have PnU(Γ)(I−Pn) = 0.This yields ηk = 0 for all k > 1. Therefore we have U(Γ) = η1I. As aconsequence, we get ∆(T ) = [η1IT − Tη1I] = 0, i.e. ∆(T ) = 0.

Now let P be an idempotent with rank(P ) = 1 and PnPPn = P forsome n.

Such an operator P is of the form P = xy∗ with Pnx = x, Pny = yand y∗x = 1, for some x, y ∈ H. Then we can find an element B ∈ B(H)such that PnBPn = B, rank(B) = n, Bx = x and dimN (B − I) = 1,where by N (B − I) it is denoted a kernel of operator B − I, i.e.N (B − I) = z : (B − I)(z) = 0. Since BP = P we get

∆(P ) = ∆(BP ) = ∆B,P (B)P +B∆B,P (P ) = B∆(P ),

because ∆B,P (B) = 0. Hence rank(∆(P )) = 1 and ∆(P ) = uv∗ for someu, v ∈ H. Then for any z ∈ H we have

0 = (B − I)∆(P )(z) = (B − I)uv∗(z) = (B − I)(v∗z)(u),

therefore u = 0, i.e. ∆(P ) = 0.

Now, let T be any operator in B(H). We choose any two vectors x, y


from the linear span of the set e1, e2, . . . , en satisfying y∗x = 1. We havealready proved that ∆(xy∗) = ∆(P ) = 0. As a consequence we get

P∆(T )P = ∆P,T (P )TP + P∆P,T (T )P + PT∆P,T (P ) = ∆P,T (PTP ) =

= (y∗Tx)∆P,T (P ) = 0

Therefore ∆(T ) = 0.

4 2-Local derivations on semi-finite real W*-algebras.

Let R be a W*-algebra, ∆ : R → R be a 2-local derivation. It easy to seethat ∆ is homogenous. Indeed, for each x ∈ R, and for λ ∈ R there exists aderivation Dx,λx such that ∆(x) = Dx,λx(x) and ∆(λx) = Dx,λx(λx). Then

∆(λx) = Dx,λx(λx) = λDx,λx(x) = λ∆(x).

Hence, ∆ is homogenous. Further, for each x ∈ R, there exists a derivationDx,x2 such that ∆(x) = Dx,x2(x) and ∆(x2) = Dx,x2(x2). Then

∆(x2) = Dx,x2(x2) = Dx,x2(x)x+ xDx,x2(x) = ∆(x)x+ x∆(x).

A linear map satisfying the above identity is called a Jordan derivation.In [9, Theorem 1] it is proved that any Jordan derivation on a semi-primealgebra is a derivation. Since every real W*-algebra R is semi-prime (i.e.aRa = 0 ⇒ a = 0), in order to prove that a 2-local derivation ∆ is aderivation it is sufficient to show that the map ∆ is additive.

Now, let R be a semi-finite W*-algebra and let τ be a faithful normalsemi-finite trace on R. Denote by mτ the definition ideal of τ , i.e. the set ofall elements x ∈ R with τ(|x|) <∞. Then mτ is a *-algebra and it is a twosided ideal of R.

It is clear that any derivation D on R maps the ideal mτ into itself.Therefore any 2-local derivation on R also maps mτ into itself.

Theorem 4.1. Let R be a finite or semi-finite W*-algebra, and let ∆ : R→R be a 2-local derivation. Then ∆ is a derivation.

Proof. By the condition of the theorem for each x ∈ R and y ∈ mτ thereexists a derivationDx,y on R such that ∆(x) = Dx,y(x) and ∆(y) = Dx,y(y).


Since every derivation on R is inner, there exists an element a ∈ R suchthat

[a, xy] = Dx,y(xy) = Dx,y(x)y + xDx,y(y) = ∆(x)y + x∆(y),

i.e. [a, xy] = ∆(x)y+ x∆(y). Hence |τ(axy)| <∞. Since mτ is an ideal andy ∈ mτ , the elements axy, xy, xya and ∆(y) also belong to mτ and hencewe have

τ(axy) = τ(a(xy)) = τ((xy)a) = τ(xya).

Thus 0 = τ(axy − xya) = τ([a, xy]) = τ(∆(x)y + x∆(y)), i.e.

τ(∆(x)y) = −τ(x∆(y)) (?)

Let a, b ∈ R be the arbitrary elements. Then in (?) replacing an element xby the sum a+ b we get

τ(∆(a+ b)y) = −τ((a+ b)∆(y)) = −τ(a∆(y))− τ(b∆(y)) =

= τ(∆(a)y) + τ(∆(b)y) = τ((∆(a) + ∆(b))y),

and so τ((∆(a + b) − ∆(a) − ∆(b))y) = 0, for all a, b ∈ R. Denote c =∆(a+ b)−∆(a)−∆(b). Then

τ(cy) = 0, ∀y ∈ mτ (??)

Now take a monotone increasing net eγγ of projections in mτ such thateγ ↑ 1I in R. Then eγc∗γ ⊂ mτ . Hence (??) implies τ(ceγc∗) = 0, ∀γ.At the same time ceγc∗ ↑ cc∗ in R. Since the trace τ is normal we haveτ(ceγc∗) ↑ τ(cc∗), i.e. τ(cc∗) = 0. The trace τ is faithful so this implies thatcc∗ = 0, i.e. c = 0. Therefore

∆(a+ b) = ∆(a) + ∆(b), a, b ∈ R,

i.e. ∆ is an additive map on R. As it was mentioned in above this impliesthat ∆ is a derivation on R.

Acknowledgement. The author would like to express his thanksto Professors Shavkat A. Ayupov and Abdugafur A. Rakhimov for theirinterest in this paper, for many helpfull remarks and corrections.

References

54

1. P.Semrl. Local automorphisms and derivations on B(H)// Proceedings ofthe American Mathematical Society. 1997. Volume 125, N9, 2677-2680.

2. S.O.Kim, J.S.Kim, Local automorphisms and derivations on Mn. Proc.Amer. Math. Soc., 132 (2004) 1389-1392].

3. Sh.A.Ayupov, K.K.Kudaybergenov. 2-local derivations and automorphismson B(H). J. Math. Anal. Appl., 395 (2012) 15-18.

4. Sh.A.Ayupov, K.K.Kudaybergenov, B.O.Nurjanov, A.K.Alauadinov. Localand 2-local derivations on noncommutative Arens algebras, Math. Slovaca,64 (2014) 423-432.

5. Sh.A.Ayupov, F.N.Arzikulov. 2-local derivations on semi-finite vonNeumann algebras, Glasgow Math. Jour. 56 (2014) 9-12.

6. Sh.A.Ayupov, K.K.Kudaybergenov. 2-local derivations on von Neumannalgebras, POSITIVITY, 19 (2015), N3, 445-455.

7. Ayupov, Sh.A., Rakhimov, A.A. and Usmanov, Sh.M.: Jordan, Real andLie Structures in Operator Algebras, Kluw.Acad.Pub., MAIA. 418, (1997),235p.

8. Ayupov, Sh.A. and Rakhimov, A.A.: Real W*-algebras, Actions of groupsand Index theory for real factors. VDM Publishing House Ltd. Beau-Bassin,Mauritius. ISBN 978-3-639-29066-0. (2010), 138p.

9. M.Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc.104 (1988), 1003-1006 .

Dadakhodjaev R.A.Institute of Mathematics, Uzbek Academy of Sciences MirzoUlugbek street, 81, 100170 Tashkent, Uzbekistan, e-mail:[email protected]

Fundamental solutions of the bi-axially ... 55

Uzbek MathematicalJournal, 2018, No1 , pp.55-64

Fundamental solutions of the bi-axially symmetricHelmholtz equation

Ergashev T.G., Hasanov A.

Abstract. The main result of this paper is the construction of fourfundamental solutions of the bi-axially symmetric multidimensional Helmholtzequation with two singular coefficients in explicit form, which could be expressedin terms of a confluent hypergeometric function of three variables. In addition, theorder of the singularity is determined and the properties of the found fundamentalsolutions that are necessary for solving boundary value problems for degenerateelliptic equations of second order are found.

Keywords: Multidimensional elliptic equation, bi-axially symmetricHelmholtz equation, fundamental solutions, confluent hypergeometric functions

Mathematics Subject Classification (2010): 33C15, 33C20, 35J15, 35J70,35J75

1 Introduction

It is known that fundamental solutions have an essential role in studyingpartial differential equations. Formulation and solving of many local andnon-local boundary value problems are based on these solutions. Moreover,fundamental solutions appear as potentials, for instance, as simple-layer anddouble-layer potentials in the theory of potentials.

The explicit form of fundamental solutions gives a possibility to studythe considered equation in detail. For example, in the works of Barros-Netoand Gelfand [1], fundamental solutions for Tricomi operator, relative to anarbitrary point in the plane were explicitly calculated. We also mentionLeray’s work [2], which it was described as a general method, based uponthe theory of analytic functions of several complex variables, for findingfundamental solutions for a class of hyperbolic linear differential operatorswith analytic coefficients. Among other results in this direction, we notea work by Itagaki [3], where 3D high-order fundamental solutions for amodified Helmholtz equation were found. The found solutions can be appliedwith the boundary particle method to some 2D inhomogeneous problems,for example, see [4].

56 Ergashev T.G., Hasanov A.

Singular partial differential equations appear at studying variousproblems of aerodynamics and gas dynamics [5] and irrigation problems[6].For instance, the famous Chaplygin equation [7] describes subsonic, sonicand supersonic flows of gas. The theory of singular partial differentialequations has many applications and possibilities of various theoreticalgeneralizations. It is, in fact, one of the rapidly developing branches ofthe theory of partial differential equations. In most cases boundary valueproblems for singular partial differential equations are based on fundamentalsolutions for these equations, for instance, see [8]. Let us consider thegeneralized bi-axially symmetric Helmholtz equation with p variables

Hp,λα,β(u) ≡

p∑i=1

∂2u

∂x2i

+2αx1

∂u

∂x1+

2βx2

∂u

∂x2− λ2u = 0 (1.1)

in the domain R+p ≡ (x1, ..., xp) : x1 > 0, x2 > 0 , where p is a dimension

of a Euclidean space (p ≥ 2), α, β and λ are constants and 0 < 2α, 2β < 1. Inthe article [9], the equation (2.1) was considered in two cases: (1) when p =2, α = 0, β > 0; (2) when p = 2, λ = 0, β > 0. In the work [10] in order to findfundamental solutions, at first two new confluent hypergeometric functionswere introduced. Furthermore, by means of the introduced hypergeometricfunction fundamental solutions of the equation (2.1) were constructed in anexplicit form. For studying the properties of the fundamental solutions, theintroduced confluent hypergeometric functions are expanded in productsby Gauss’s hypergeometric functions. The logarithmic singularity of theconstructed fundamental solutions of equation (2.1) was explored with thehelp of the obtained expansion.Fundamental solutions of equation (2.1) withp = 3 and λ = 0 were used in the investigation of the Dirichlet problem forthree-dimensional elliptic equation with two singular coefficients [11].

In the present article for the equation (2.1) in the domain R+p at p > 2

four fundamental solutions are constructed in explicit form. Furthermore,some properties of these solutions are shown, which will be used for solvingboundary value problems for aforementioned equation.


2 About one confluent hypergeometricfunction of three variables

The confluent hypergeometric function of three variables which we will usein the present work looks like [10]

A2(a; b1, b2; c1, c2;x, y, z) =∞∑

m,n,k=0

(a)m+n−k(b1)m(b2)n(c1)m(c2)nm!n!k! xmynzk, (2.1)

where a, b1, b2, c1, c2 are complex constants, c1, c2 6= 0,−1,−2, ... and(a)n = Γ(a+ n)/Γ(a) is the Pochhammer symbol.

Using the formula of derivation

∂i+j+k

∂xi∂yj∂zkA2(a; b1, b2; c1, c2;x, y, z)

=(a)i+j−k(b1)i(b2)j

(c1)i(c2)jA2(a+i+j−k; b1+i, b2+j; c1+i, c2+j;x, y, z), (2.2)

it is easy to show that the hypergeometric function A2(a; b1, b2; c1, c2;x, y, z)satisfies the system of hypergeometric equations [10]

x(1− x)ωxx − xyωxy + xzωxz + [c1 − (a+ b1 + 1)x]ωx−b1yωy + b1zωz − ab1ω = 0,

y(1− y)ωyy − xyωxy + yzωyz + [c2 − (a+ b2 + 1)y]ωy−b2xωx + b2zωz − ab2ω = 0,

zωzz − xωxz − yωyz + (1− a)ωz + ω = 0,

(2.3)

whereω(x, y, z) = A2(a; b1, b2; c1, c2;x, y, z). (2.4)

Really, by virtue of the derivation formula (2.2), it is easy to calculatethe following expressions

ωx =∞∑

m,n,k=0

(a)m+n−k(b1)m(b2)n(c1)m(c2)nm!n!k!

(a+m+ n− k)(b1 +m)(c1 +m)

xmynzk, (2.5)

xωx =∞∑

m,n,k=0


m

1xmynzk, (2.6)


yωy =∞∑

m,n,k=0


n

1xmynzk, (2.7)

zωz =∞∑

m,n,k=0


k

1xmynzk, (2.8)

xyωxy =∞∑

m,n,k=0


mn

1xmynzk, (2.9)

xzωxz =∞∑

m,n,k=0


mk

1xmynzk, (2.10)

x2ωxx =∞∑

m,n,k=0


(m− 1)m1

xmynzk. (2.11)

Substituting (2.5)-(2.11) into the first equation of the system (2.3), weare convinced that function ω(x, y, z) satisfies this equation. We are similarlyconvinced that function ω(x, y, z) satisfies the second and third equationsof the system (2.3).

Having substituted ω = xτyνzµψ(x, y, z) in the system (2.3), it ispossible to be convinced that for the values

τ : 0, 1− c1, 0, 1− c1,ν : 0, 0, 1− c2, 1− c2,µ : 0, 0, 0, 0,

the system has four linearly independent solutions

ω1(x, y, z) = A2(a; b1, b2; c1, c2;x, y, z), (2.12)

ω2(x, y, z) = x1−c1A2(a+ 1− c1; b1 + 1− c1, b2; 2− c1, c2;x, y, z), (2.13)

ω3(x, y, z) = y1−c2A2(a+ 1− c2; b1, b2 + 1− c2; c1, 2− c2;x, y, z), (2.14)

ω4(x, y, z) = x1−c1y1−c2

×A2(a+ 2− c1 − c2; b1 + 1− c1, b2 + 1− c2; 2− c1, 2− c2;x, y, z). (2.15)


In order to further study the decomposition properties of the productsby Gauss’s hypergeometric functions, we need to know the same expansionsof the function A2(a; b1, b2; c1, c2;x, y, z). For this purpose we shall considerthe expression

A2(a; b1, b2; c1, c2;x, y, z)

=∞∑k=0

(−z)k

(1− a)kk!F2(a− k; b1, b2; c1, c2;x, y), (2.16)

where

F2(a; b1, b2; c1, c2;x, y) =∞∑

m,n=0

(a)m+n(b1)m(b2)n(c1)m(c2)nm!n!

xmyn.

In [12] for Appell’s hypergeometric function F2(a; b1, b2; c1, c2;x, y)following expansion was found

F2(a; b1, b2; c1, c2;x, y) =∞∑i=0

(a)i(b1)i(b2)i(c1)i(c2)ii!

xiyi×

×F (a+ i; b1 + i; c1 + i;x)F (a+ i; b2 + i; c2 + i; y), (2.17)

where F (a, b; c; z) =∞∑n=0

(a)n(b)n(c)nn! z

n is a hypergeometric function of Gauss.

Considering expansion (2.17), from the identity (2.16) we find [10]

A2(a; b1, b2; c1, c2;x, y, z) =∞∑

i,j=0

(a)i−j(b1)i(b2)i(c1)i(c2)ii!j!

xiyizj×

×F (a+ i− j, b1 + i; c1 + i;x)F (a+ i− j, b2 + i; c2 + i; y). (2.18)

By virtue of the formula

F (a, b; c;x) = (1− x)−bF(c− a, b; c;

x

x− 1

),


we get from expansion (2.18)

A2(a; b1, b2; c1, c2;x, y, z) = (1− x)−b1(1− y)−b2×

×∞∑

i,j=0

(a)i−j(b1)i(b2)i(c1)i(c2)ii!j!

(x

1−x

)i (y

1−y

)izj×

×F(c1 − a+ j, b1 + i; c1 + i; x

x−1

)×

×F(c2 − a+ j, b2 + i; c2 + i; y

y−1

).

(2.19)

Expansion (2.19) will be used for studying properties of the fundamentalsolutions.

We note, that the expansions for the hypergeometric function ofLauricella F (s)

A were found in [13].

3 Fundamental solutionsWe consider the generalized bi-axially symmetric multidimensionalHelmholtz equation in the domain R+

p . The equation (2.1) has the followingconstructive formulas

Hp,λα,β

(x1−2α

1 u)≡ x1−2α

1 Hp,λ1−α,β(u), (3.1)

Hp,λα,β

(x1−2β

2 u)≡ x1−2β

2 Hp,λα,1−β(u). (3.2)

The constructive formulas (3.1) and (3.2) give the possibility to solveboundary value problems for equation (2.1) for various values of theparameters α, β.

We search the solution of equation (2.1) in the form

u(x) = P (r)ω(ξ, η, ζ), (3.3)

where

r2 =p∑i=1

(xi − x0i)2, r21 = (x1 + x01)

2 +p∑i=2

(xi − x0i)2,

r22 = (x1 − x01)2 + (x2 + x02)

2 +p∑i=3

(xi − x0i)2,


ξ =r2 − r21r2

= −4x1x01

r2, η =

r2 − r22r2

= −4x2x02

r2,

ζ = −λ2

4r2, P (r) =

(r2)1−α−β− p

2 .

Substituting (3.3) into equation (2.1), we have

A1ωξξ +A2ωηη +A3ωζζ +B1ωξη +B2ωξζ +B3ωηζ

+C1ωξ + C2ωη + C3ωζ +Dω = 0, (3.4)

where

A1 = P

p∑i=1

(∂ξ

∂xi

)2

, A2 = P

p∑i=1

(∂η

∂xi

)2

, A3 = P

p∑i=1

(∂ζ

∂xi

)2

,

B1 = 2Pp∑i=1

∂ξ

∂xi

∂η

∂xi, B2 = 2P

p∑i=1

∂ξ

∂xi

∂ζ

∂xi, B3 = 2P

p∑i=1

∂η

∂xi

∂ζ

∂xi,

C1 = 2p∑i=1

∂P

∂xi

∂ξ

∂xi+ P

p∑i=1

∂2ξ

∂x2i

+ P

(2αx1

∂ξ

∂x1+

2βx2

∂ξ

∂x2

),

C2 = 2p∑i=1

∂P

∂xi

∂η

∂xi+ P

p∑i=1

∂2η

∂x2i

+ P

(2αx1

∂η

∂x1+

2βx2

∂η

∂x2

),

C3 = 2p∑i=1

∂P

∂xi

∂ζ

∂xi+ P

p∑i=1

∂2ζ

∂x2i

+ P

(2αx1

∂ζ

∂x1+

2βx2

∂ζ

∂x2

),

D =p∑i=1

∂2P

∂x2i

+2αx1

∂P

∂x1+

2βx2

∂P

∂x2− λ2P.

After elementary evaluations we find

A1 = −4Pr2

x01

x1ξ(1− ξ), A2 = −4P

r2x02

x2η(1− η), (3.5)

A3 = −λ2Pζ, B1 =4Pr2

x01

x1ξη +

4Pr2

x02

x2ξη, (3.6)

B2 = −4Pr2

x01

x1ξζ + λ2Pξ, B3 = −4P

r2x02

x2ηζ + λ2Pη, (3.7)


C1 = −4Pr2

x01

x1

[2α−

(2α+ β +

p

2

)ξ]

+4Pr2

x02

x2βξ, (3.8)

C2 =4Pr2

x01

x1αη − 4P

r2x02

x2

[2β −

(α+ 2β +

p

2

)η], (3.9)

C3 = −4Pr2

x01

x1αζ − 4P

r2x02

x2βζ − λ2P

(p2− α− β

), (3.10)

D =4Pr2

[x01

x1α+

x02

x2β

](α+ β − 1 +

p

2

)− λ2P. (3.11)

Substituting equalities (3.5)-(3.11) into equation (3.4), we get the system ofhypergeometric equations

ξ(1− ξ)ωξξ − ξηωξη + ξζωξζ +[2α−

(2α+ β + p

2

)ξ]ωξ

−αηωη + αζωζ − α(α+ β − 1 + p

2

)ω = 0,

η(1− η)ωηη − ξηωξη + ηζωηζ +[2β −

(α+ 2β + p

2

)η]ωη

−βξωξ + βζωζ − β(α+ β − 1 + p

2

)ω = 0,

ζωζζ − ξωξζ − ηωηζ +(2− α− β − p

2

)ωζ + ω = 0.

(3.12)

Considering the solutions of the system of hypergeometric equations(2.12)-(2.15), we define

ω1(ξ, η, ζ) = A2

(α+ β − 1 +

p

2;α, β; 2α, 2β; ξ, η, ζ

), (3.13)

ω2(ξ, η, ζ) = ξ1−2αA2

(−α+ β +

p

2; 1− α, β; 2− 2α, 2β; ξ, η, ζ

), (3.14)

ω3(ξ, η, ζ) = η1−2βA2

(α− β +

p

2;α, 1− β; 2α, 2− 2β; ξ, η, ζ

), (3.15)

ω4(ξ, η, ζ) = ξ1−2αη1−2β××A2

(1− α− β + p

2 ; 1− α, 1− β; 2− 2α, 2− 2β; ξ, η, ζ).

(3.16)

Substituting the equalities (3.13)-(3.16) into the expression (3.3), we getsome solutions of the equation (2.1)q1(x, x0) = k1

(r2)1−α−β− p

2 ×

×A2

(α+ β − 1 +

p

2;α, β; 2α, 2β; ξ, η, ζ

), (3.17)


q2(x, x0) = k2

(r2)α−β− p

2 x1−2α1 x1−2α

01 ×

×A2

(−α+ β +

p

2; 1− α, β; 2− 2α, 2β; ξ, η, ζ

), (3.18)

q3(x, x0) = k3

(r2)−α+β− p

2 x1−2β2 x1−2β

02 ×

×A2

(α− β +

p

2;α, 1− β; 2α, 2− 2β; ξ, η, ζ

), (3.19)

q4(x, x0) = k4

(r2)−1+α+β− p

2 x1−2α1 x1−2α

01 x1−2β2 x1−2β

02 ×

×A2

(1− α− β +

p

2; 1− α, 1− β; 2− 2α, 2− 2β; ξ, η, ζ

), (3.20)

where k1, ...k4 are constants which will be determined at solving boundaryvalue problems for equation (2.1). It is easy to notice that the consideredfunctions (3.17)-(3.20) possess the properties

∂q1(x, x0)∂x1

|x1=0= 0,∂q1(x, x0)

∂x2|x2=0= 0,

q2(x, x0) |x1=0= 0,∂q2(x, x0)

∂x2|x2=0= 0,

∂q3(x, x0)∂x1

|x1=0= 0, q3(x, x0) |x2=0= 0,

q4(x, x0) |x1=0= 0, q4(x, x0) |x2=0= 0.

From the expansion (2.19) follows that the fundamental solutions (3.17)-(3.20) at r → 0 possess a singularity of the order 1

rp−2 , where p > 2.

References

1. J.J. Barros-Neto and I.M.Gelfand, Fundamental solutions for the Tricomioperator, I,II,III, Duke Math.J. 98,3(1999), 465–483; 111,3(2001), 561–584;128,1(2005), 119–140.

2. J.Leray, Un prolongementa de la transformation de Laplace qui transformela solution unitaires d’un opereteur hyperbolique en sa solution elementaire(probleme de Cauchy,IV), Bull.Soc.Math.France, 90( 1962), 39–156.

3. M.Itagaki, Higher order three-dimensional fundamental solutions to theHelmholtz and the modified Helmholtz equations, Eng. Anal. Bound. Elem.,15(1995), 289–293.


4. M.A.Golberg and C.S.Chen, The method of fundamental solutionsfor potential, Helmholtz and diffusion problems, in: Golberg M.A.(Ed.),Boundary Integral Methods-Numerical and Mathematical Aspects,Comput.Mech.Publ.,1998, 103-176.

5. L.Bers, Mathematical aspects of subsonic and transonic gas dynamics, NewYork,London. 1958.

6. L.I.Serbina, A problem for the linearized Boussinesq equation with anonlocal Samarskii condition, Differential Equations, 38,8 (2002), 1187–1194.

7. S.A.Chapligin, On gas streams, Dissertation, Moscow, 1902 (in Russian).

8. M.S.Salakhitdinov and A.Hasanov, A solution of the Neumann-Dirichletboundary-value problem for generalized bi-axially symmetric Helmholtzequation Complex Variables and Elliptic Equations, 53, 4(2008), 355–364.

9. O.I. Marichev, Integral representation of solutions of the generalizeddouble axial-symmetric Helmholtz equation (in Russian), Differencial’nyeUravneniya, Minsk, 14,10(1978), 1824–1831.

10. A.Hasanov, Fundamental solutions of the generalized bi-axially symmetricHelmholtz equation, Complex Variables and Elliptic Equations, 52, 8( 2007),673–683.

11. E.T.Karimov and J.J.Nieto, The Dirichlet problem for a 3D ellipticequation with two singular coefficients, Computers and Mathematics withApplications. 62(2011), 214–224.

12. J.L.Burchnall and T.W.Chaundy, Expansions of Appell’s doublehypergeometric functions, The Quarterly Journal of Mathematics,Oxford, Ser.12(1941),112–128.

13. A.Hasanov and H.M.Srivastava, Some decomposition formulas associatedwith the Lauricella function F r

A and other multiple hypergeometric functions,Applied Mathematics Letters, 19,2(2006),113–121.

Ergashev T.G.Institute of Mathematics, Mirzo Ulugbek street, 81, 100170,Tashkent, Uzbekistan, e-mail: [email protected] A.Institute of Mathematics, Mirzo Ulugbek street, 81, 100170,Tashkent, Uzbekistan, e-mail: [email protected]

A differential operator of order 2m and its fundamental solution 65


A differential operator of order 2m and itsfundamental solution

Jalolov I.I.

Abstract. In this paper we consider a construction of a differential operatorof order 2m general form and proved that Vm (x) is a fundamental solution of thedifferential operator.

Keywords: differential operator, discrete analogueMathematics Subject Classification (2010):

1 Introduction.When studying the various issues arising in the theory of approximateintegration of differential and partial differential equations, it is provedvery fruitful so-called functional approach. The essence of this approach (ifit is limited to an example of a boundary value problem for a differentialequation) is that the differential equation, together with the boundaryconditions, is realized as an operator acting in a specially selected functionalspace, the required information is extracted from the properties of thisoperator. When solving many problems in the theory of approximateintegration and differential equations, the right choice of spaces was thekey to success. A very well-chosen approach was demonstrated in thewell-known papers on the poly-harmonic equation of S.L. Sobolev (see [1,2]). In these papers, the space W (`)

p (Ω) of functions was associated with theoperator - ∆`, all derivatives of which up to order ` are summable in the p-th power over the domain Ω. It turned out that under certain conditions thefunctions in the space W (`)

p (Ω) have some regularity properties on surfacesof dimension less than n. This allowed S.L. Sobolev to supply and solveby the variational method the first boundary-value problem for equation∆ù = f with boundary conditions on surfaces of various dimensions. Todescribe an analytic algorithm for finding the optimal coefficients, S.L.Sobolev defined and investigated the discrete analogue D

(m)hH [β] of the

polyharmonic operator ∆m [1]. The problem of constructing a discreteoperator D(m)

hH [β] for an arbitrary operator turned out to be very difficult.In the one-dimensional case Z. Zh. Zhamolov and Kh. M. Shadimetovwere engaged in the construction of the discrete analogue D(m)

h [β] of the

66 Jalolov I.I.

differential operator d2m

dx2m [3,4]. The construction of a discrete analogueof the differential operator d2m

dx2m − d2m−2

dx2m−2 and d4/dx4 + 2d2

/dx2 + 1 was

carried out by Kh.M. Shadimetov and A.R. Hayotov [5, 6].

2 Statement of the problem.

The work of many authors is devoted to the construction of optimalquadrature and cubature formulas by the method proposed byS.L. Sobolev(see [3] - [13]). On the basis of this method, an algorithm for finding theoptimal coefficients of weighted quadrature formulas in the S.L. SobolevspaceW (m)

2 (R) was developed in [14] - [16] and the existence and uniquenessof these formulas are proved. And also, we consider the problem ofconstructing a discrete function Dm [β] with m = 1, 4, which satisfiesequality

Dm [β] ∗ νm [β] = δ [β] , (2.1)

where

νm(x) =π · e−2π|x|

22m−2(m− 1)!

m−1∑k=0

(2m− k − 2)!(4π)k

k!(m− k − 1)!|x|k , (see.[17]) (2.2)

δ[β] = 1, β = 0; 0, β 6= 0, h = 1N , N = 1, 2, ... In the following papers

(see [3, 4, 6, 7]), discrete analogues of certain differential operators areconstructed that have an important role in calculating the coefficients ofoptimal quadrature, cubature formulas and optimal interpolation formulas.In [14-16], discrete analogues of one differential operator with the help ofthe function νm (x) satisfying equality (2.1) were constructed, but here theform of the differential operator was not known to us. Because of this, inthis paper we consider the problem of constructing a differential operatorof order 2m of the general form and prove that Vm (x) is a fundamentalsolution of this differential operator. Here

Vm (x) = ωνm (ωx) , (2.3)

where ω > 0 and substituting ωx in the place x, from (2.2) we obtain theform of the function νm (ωx).


3 Construction of a differential operator ofthe 2m−th order and finding a fundamentalsolution

Since the discrete functionDm [β] is used in the calculation of the coefficientsof optimal quadrature formulas in the space of S.L. Sobolev Wm

2 (R), thenit is expedient to determine this space. Definition. The space Wm

2 (R) isdefined as the closure of infinitely differentiable functions defined in R anddecreasing to infinity faster than any negative degree in the norm (see [2,17])

‖f(x)|Wm2 (R)‖ =

∞∫

−∞

|F−1[(1 + y2)m2 F [f(x)](y)]|2dx

12

.

Here R and F−1 are the direct and the inverse Fourier transforms:

F [f(x)](y) =

∞∫−∞

f(x)e2πiyxdx

and

F−1[f(x)](y) =

∞∫−∞

f(x)e−2πiyxdx.

Taking (2.3) into account, equalities (2.1) and (2.2) we have the followingform:

Dm [β] ∗ Vm [β] = δ [β] , (3.1)

where

Vm(x) =πωe−2πω|x|

22m−2(m− 1)!

m−1∑k=0

(2m− k − 2)!(4πω)k

k!(m− k − 1)!|x|k , (3.2)

δ[β] = 1, β = 0; 0, β 6= 0, h = 1N , ω > 0, N = 1, 2, ... Note that equation (3.1)

is a discrete analog of the following equation(a− b

d2

dx2

)mVm (x) = δ (x) , (3.3)

68 Jalolov I.I.

where Vm (x) is defined by (2.2), δ (x) is the Dirac delta- function, a and bare unknowns. We determine the values of a and b from (3.3) by induction.Let m = 1, then equation (3.3) go to the next(

a− bd2

dx2

)V1 (x) = δ (x) , (3.4)

that is, we must find a value of a and b such that equality (3.3) is satisfied.For m = 1 from (3.2) we have

V1 (x) = πωe−2πω |x|. (3.5)

The generalized derivatives V′

1 (x) and V′′

1 (x) have the form

V′

1 (x) = −2(πω)2e−2πω |x|sign (x) ,

V′′

1 (x) = 4(πω)3e−2πω |x| − 4(πω)2δ (x) . (3.6)

Using (3.6) from (3.4) we obtain a = 1 and b = 14(πω)2

. Thus, for a = 1 andb = 1

4(πω)2(3.4) has the following form(

1− d2

(2πω )2dx2

)V1 (x) = δ (x) . (3.7)

Thus, we have found a differential operator(1− d2

(2πω)2dx2

), (3.8)

which, function D1 [β] is a discrete analogue of (3.8), and satisfies (2.1) form = 1. It follows that from (3.7) V1 (x) is a fundamental solution of thedifferential operator (3.8). Now let m = 2, then from (3.2) we obtain

V2 (x) =πω

2e−2πω |x| (1 + 2πω |x|) . (3.9)

We show that V2 (x) is a solution of an equation of the form(1− d2

(2πω )2dx2

)2

V2 (x) = δ (x) , (3.10)


or (1− d2

(2πω )2dx2

)(1− d2

(2πω )2dx2

)V2 (x) = δ (x) . (3.11)

Computing the generalized derivatives of V′

2 (x) and V′′

2 (x), we have

V′

2 (x) = −(πω)3e−2πω|x| |x| signx,

V′′

2 (x) = −2(πω)3[`−2πω |x| (−2πω |x|+ 1)

]. (3.12)

Using (3.12) from (3.11), we have(1− d2

(2πω )2dx2

)V2 (x) =

πω

2e−2πω |x| (2πω |x|+ 1)+

+πω

2e−2πω|x| (−2πω |x|+ 1)πωe−2πω |x|. (3.13)

Hence, taking into account (3.9) and (3.10) from (3.11), we obtain(1− d2

(2πω)2dx2

)2

V2 (x) =

(1− d2

(2πω)2dx2

)(1− d2

(2πω)2dx2

)V2 (x) =

=

(1− d2

(2πω)2dx2

)πωe−2πω |x| = δ (x) . (3.14)

Thus we have proved equality (3.10). It follows that we have found adifferential operator (

1− d2

(2πω)2dx2

)2

, (3.15)

which is a function D2 [β] is a discrete analogue of the operator (3.15),satisfying equality (2.1) with m = 2. Taking into account the parity of thefunction Vm (x), that is, Vm (x) = Vm (−x) we have

Vm (x) =Vm (x) + Vm (−x)

2=

12ω

∞∫−∞

e2πiyωx + e−2πiyωx(1 + (y)2

)m dy =

=12ω

∞∫−∞

2 cos (2πyωx)(1 + (y)2

)m dy=ω

∞∫−∞

cos (2πyωx)(1 + (y)2

)m dy, (3.16)

70 Jalolov I.I.

or

Vm (x) = 2ω

∞∫0

cos (2πyωx)(1 + (y)2

)m dy. (3.17)

In order to get the differential operator(1− d2

(2πω)2dx2

)m, we need the

following:

Lemma 3.1. The following recurrence formula holds(1− d2

(2πω)2dx2

)Vm (x) = Vm−1 (x) ,m = 2, 3, ..., ν1 (x) = πωe−2πω|x|.

(3.18)

Proof. Computing the derivative of a function of the form (3.17), wehave

V′

m (x) = 2ω

∞∫0

−2πωy sin (2πωyx)(1 + (y)2

)m dy,

V′′

m (x) = 2ω

∞∫0

−(2πωy)2 cos (2πωyx)(1 + (y)2

)m dy. (3.19)

Now applying (3.19) to the left-hand side of (3.18), we obtain(1− d2

(2πω)2dx2

)Vm (x) = 2ω

∞∫0

cos (2πωyx)(1 + (y)2

)m dy+

+2(2πωy)2

(2πω)2ω

∞∫0


)m dy = 2ω

∞∫0


)(1 + (y)2

)m dy =

2ω

∞∫0


)m−1 dy = Vm−1 (x) , (3.20)

which was proved.Thus we have proved the following


Theorem 3.2. The differential operator of the 2m-th order, which functionDm [β] is the discrete analogue of this operator, satisfying the equality (2.1)has the following form of (

1− d2

(2πω)2dx2

)m, (3.21)

and the function Vm (x) is a fundamental solution of the differential operator(3.21), that is, the following equality holds(

1− d2

(2πω)2dx2

)mVm (x) = δ (x) . (3.22)

At the end, we note that the constructed differential operator has numerousapplications in mechanics, especially in problems of the theory of oscillations(see [18]).

References1. S.L. Sobolev. Some applications of functional analysis in mathematical

physics.-Moscow: Nauka, 1988. - 333 p.

2. Sobolev S.L. Introduction to the theory of cubature formulas. Moscow:Nauka, 1974. - 808 p.

3. Zhamolov Z.Zh. On a difference analogue of the operator d2m

dx2m andits construction. In: Direct and inverse problems for partial differentialequations and their applications. Tashkent. Fan, 1978, -pp. 97 - 108.

4. Kh. M. Shadimetov. Discrete analogue of differential operator d2m

dx2m and itsstructure. Questions of computational and applied mathematics. - Tashkent,1985, -pp. 22-25.

5. Kh. M. Shadimetov. On the calculation of the coefficients of optimalquadrature formulas. DAN USSR, 1980, No. 4, -pp. 4-7.

6. Shadimetov Kh.M., Hayotov A.R. Construction of a discrete analogue of adifferential operator d2m

dx2m − d2m−2

dx2m−2 . Uzbek Matematical Journal, 2004, no.2, -pp. 85-95.

7. Hayotov A.R. Construction of a discrete analogue of the differentialoperator d4

dx4 + 2 d2

dx2 + 1 and its properties. Uzbek Matematical Journal,2009, є3, -pp. 81-88.

8. Zhamolov Z.Zh., Solikhov G.N, Sharipov T.H, Approximate integrationof smooth functions, Mathematical analysis and related questions ofmathematics, Nauka, Sibirsk. Otd., Novosibirsk, 1978. - pp. 37-60.

72 Jalolov I.I.

9. Ramazanov M.D, Shadimetov Kh.M. Calculation of the coefficients ofoptimal quadrature formulas in space W

(m,m−1)2 // Uzbek Mathematical

Journal. -Tashkent, 2004.-є 3. -pp. 80-98.

10. Shadimetov Kh.M., Hayotov A.R. Calculation of the coefficients of optimalquadrature formulas in space W (m,m−1)

2 // Uzbek Mathematical Journal.-Tashkent, 2004.-є 3. -pp. 80-98.

11. Shadimetov Kh.M, Hayotov A.R. Optimal quadrature formulas withpositive coefficients in L

(m)2 (0, 1). Journal of Computational and Applied

Mathematics 235 (2011) 1114-1128.

12. Hayotov A.R, Milovanovi G.V, Shadimetov Kh.M. On an optimalquadrature formula in the sense of Sard. Numerical Algorithms, 2011, vol.57, pp. 487-510.

13. Shoinzhurov Ts.B. Linear functionals in the space of Sobolev. Ulan-Ude:Publishing house of the All-Russian State Technical University of SB RAS,2009. PP.214-224.

14. Shadimetov Kh.M., Zhalolov I.I. On an algorithm for constructing anoperator for determining the optimal coefficients of weighted quadratureformulas in space. - Uzbek Mathematical Journal 2010, є3, -pp. 178-187.

15. Shadimetov Kh. M., Zhalolov Ik. I. On an algorithm for constructing adiscrete analogue D2 [β] of a single operator. - Uzbek Mathematical Journal2015, No.1, -s. 158-163.

16. Shadimetov Kh. M., Zhalolov Ik. I. Algorithm for constructing a discreteanalogue D4 [β] of a single operator. Proceedings of the InternationalConference "Actual Problems of Applied Mathematics and InformationTechnologies-Al Khorezmi 2016 T1-C.25-26.

17. Gradshtein I.S, Ryzhik I.I. Tables of works. integrals, sums of series, andScience, fiz. Mat., 1971.

18. V. Kech, P. Theodorescu. Introduction to the theory of generalizedfunctions with applications in engineering.-M .: Mir, 1978. 518 p.

Jalolov I.I.Teacher of department of Computer graphics and informationtechnology, Engineers of Railway Tashkent Institute

On solvability of the mixed problem for a partial equation ... 73


On solvability of the mixed problem for a partialequation of a fractional order with Laplace operators

and nonlocal boundary conditions in the Sobolevclasses

Kasimov Sh. G., Ataev Sh. K.

Abstract. In this paper we study on solvability of mixed problem for partialdifferential equation fractional order with Laplas operator with nonlocal boundaryconditions. On the completeness property of the eigenfunction system of the Laplasoperator with nonlocal boundary conditions.

Keywords: Banach Space; Sobolev Space;Laplace operators; nonlocalboundary conditions

Mathematics Subject Classification (2010): 35K20, 35K51, 35K58

1 Introduction.

As it is known, so-called fractal media are studied in solid state physics,in particular, the diffusion phenomena in them. In one of the models,the diffusion in a strongly porous medium is described by the heatconduction equation, but with a fractional derivative with respect to thetime coordinate. In the present work we consider the equation of the form

Dα0tu(x, t) = ∆u(x, t) + f(x, t), (x, t) ∈ Π× (0,∞), l − 1 < α ≤ l, l ∈ N

(1.1)with initial conditions

limt→0

Dα−k0t u(x, t) = ϕk(x), k = 1, 2, ..., l (1.2)

74 Kasimov Sh.G., Ataev Sh.K.

and boundary conditions

αj · u(x1, ..., xj , ..., xN , t) |xj=0 +βj · u(x1, ..., xj , ..., xN , t) |xj=π= 0,1 ≤ j ≤ p,

βj · ∂u(x1,...,xj ,...,xN ,t)∂xj

|xj=0 +αj · ∂u(x1,...,xj ,...,xN ,t)∂xj

|xj=π= 0,1 ≤ j ≤ p,u(x1, ..., xj , ..., xN , t) |xj=0= u(x1, ..., xj , ..., xN , t) |xj=π,p+ 1 ≤ j ≤ q,∂u(x1,...,xj ,...,xN ,t)

∂xj|xj=0=

∂u(x1,...,xj ,...,xN ,t)∂xj

|xj=π,p+ 1 ≤ j ≤ q,u(x1, ..., xj , ..., xN , t) |xj=0= 0, q + 1 ≤ j ≤ N,u(x1, ..., xj , ..., xN , t) |xj=π= 0, q + 1 ≤ j ≤ N,1 ≤ p ≤ q ≤ N,

(1.3)where (x, t) = (x1, ..., xj , ..., xN , t) ∈ Π × (0,∞), Π = (0, π) × ... × (0, π),αj = const, βj = const and f(x, t), ϕk(x), k = 1, 2, ..., l., arefunctions that can be expanded in terms of the system of eigenfunctionsvn(x), n ∈ ZN of the spectral problem:

∆v(x) + µv(x) = 0, (1.4)

αj · v(x1, ..., xj , ..., xN ) |xj=0 +βj · v(x1, ..., xj , ..., xN ) |xj=π= 0,1 ≤ j ≤ p,

βj · ∂v(x1,...,xj ,...,xN )∂xj

|xj=0 +αj · ∂v(x1,...,xj ,...,xN )∂xj

|xj=π= 0,1 ≤ j ≤ p,v(x1, ..., xj , ..., xN ) |xj=0= v(x1, ..., xj , ..., xN ) |xj=π,p+ 1 ≤ j ≤ q,∂v(x1,...,xj ,...,xN )

∂xj|xj=0=

∂v(x1,...,xj ,...,xN )∂xj

|xj=π,p+ 1 ≤ j ≤ q,v(x1, ..., xj , ..., xN ) |xj=0= 0, q + 1 ≤ j ≤ N,v(x1, ..., xj , ..., xN ) |xj=π= 0, q + 1 ≤ j ≤ N,1 ≤ p ≤ q ≤ N,

(1.5)Here for α < 0, the fractional integral Dα has the form

Dαatu(x, t) =

sign(t− a)Γ(−α)

t∫a

u(x, τ) · dτ|t− τ |α+1 ,


Dαatu(x, t) = u(x, t) for α = 0, and for l − 1 < α ≤ l, l ∈ N, the fractional

derivative has the form

Dαatu(x, t) = signl(t− a)

dl

dtlDα−lat u(x, t) =

=signl+1(t− a)

Γ(l − α)dl

dtl

t∫a

u(x, τ) · dτ|t− τ |α−l+1

.

We look for eigenfunctions of the spectral problem (1.4)-(1.5) in the formof the product v(x) = y1(x1) · · · · · yN (xN ). Then we obtain instead of thespectral problem (1.4)-(1.5) the following spectral problem

−y′′(x) = µy(x), µ = λ2 (1.6)αy(0) + βy(π) = 0βy′(0) + αy′(π) = 0 (1.7)

In the case of |α| = |β|, i.e. with boundary conditions y(0) = y(π),y′(0) = y′(π) or y(0) = −y(π), y′(0) = −y′(π), the spectral problem (1.6)-(1.7) was investigated by many authors (see, for example, [1]-[6],[8]-[9]). Inorder to simplify calculations, we confine ourselves to the case of |α| 6= |β|,α 6= 0, β 6= 0. It is not difficult to see that µ = 0 is not an eigenvalue ofthe problem (1.6)-(1.7). In fact, if µ = 0 is the eigenvalue, then y′′ = 0,y = ax + b, αb + β(aπ + b) = 0, βa + αa = 0. We obtain from here a = 0,b = 0, i.e. y ≡ 0. Similarly, for µ < 0 the problem (1.6)-(1.7) has notnontrivial solutions.

For µ > 0, the general solution of (1.6) has the form

y(x) = Acosλx+Bsinλx. (1.8)

We have from boundary conditions:

αy(0) + βy(π) = αA+ β(Acosλπ +Bsinλπ) = 0,

βy′(0) + αy′(π) = β(λB) + α(λBcosλπ − λAsinλπ) = 0,

i.e. (α+ βcosλπ)A+ βsinλπB = 0αsinλπA− (β + αcosλπ)B = 0 (1.9)

Hence, nontrivial solutions of (1.6)-(1.7) are possible only in the case of

(α+ βcosλπ)(−β − αcosλπ)− αβsin2(λπ) = 0. (1.10)


Furthermore,

−αβ − α2cosλπ − β2cosλπ − αβcos2λπ − αβsin2λπ = 0,

i.e. −(α2 + β2)cosλπ = 2αβ or cosλπ =−2αβα2 + β2

. Therefore

λπ = arccos−2αβα2 + β2

or

λπ = ±arccos −2αβα2 + β2

+ 2nπ, n = 1, 2, ... . (1.11)

Further,µ±n = (2n+ εnϕ)2 = (−2n− εnϕ)2 = µ∓−n,

εn = ±1, ϕ =1π

arccos−2αβα2 + β2

, n ∈ Z.

That’s why µ±n 6= µ±−n means that ε−n 6= −εn, i.e. ε−n = εn, n ∈ Z. Thus,eigenvalues and eigenfunctions of (1.6)-(1.7) are

µn = λ2n = (2n+ εnϕ)2, ϕ =

1πarccos

−2αβα2 + β2

, εn = ±1, ε−n = εn, n ∈ Z

(1.12)and

yn(x) = Bn

(β + αcosλnπ

αsinλnπcosλnx+ sinλnx

), (1.13)

respectively, where

β + α cosλnπα sinλnπ

=β − 2α2β

α2+β2

εnα√

1− 4α2β2

(α2+β2)2

=

=β(β2 − α2)

εnα | β2 − α2 |= εnsign(β2 − α2)

β

α

and, hence, yn(x) = Bn

(εnsign(β2 − α2)

β

αcosλnx+ sinλnx

). Choosing

Bn = εnsign(β2 − α2)α√

α2 + β2

√2π

1√1 + (2n)2s

we obtain


yn(x) =

√2π

1√α2 + β2

1√1 + (2n)2s

·

·(β cosλnx+ εnsign(β2 − α2)α sinλnx).

Denote

ωn =√

2π

1√α2+β2

1√1+(2n)2s

. Then

yn(x) = ωn(βcosλnx+ εnsign(β2 − α2)αsinλnx). (1.14)

The norm in the space W s2 (0, π) is introduced as follows:∥∥∥f∥∥∥2

W s2 (0,π)

=∥∥∥f∥∥∥2

L2(0,π)+∥∥∥Dsf

∥∥∥2

L2(0,π).

Let εn = ε−n. Then the system of vectors

zn(x) = ωn(β cos 2nx+ εnsign(β2 − α2)α sin 2nx

)forms the complete orthonormal system in W s

2 (0, π).The following lemmas hold.

Lemma 1.1. Let an be a finite system of complex numbers. Then thefollowing inequalities∥∥∥∥∥

N∑−N

an(yn(x)− zn(x))

∥∥∥∥∥L2(0,π)

≤√

2 · maxx∈[0,π]

∣∣∣eiϕx − 1∣∣∣ ·√√√√ N∑

−N| an · cn |2

are valid where

cn =1√

1 + (2n)2s. (1.15)

Lemma 1.2. Let an be a finite system of complex numbers. Then thefollowing inequalities

∥∥∥DsN∑−N

an(yn(x)− zn(x))∥∥∥L2(0,π)

≤


≤√

2[

maxx∈[0,π]

∣∣∣eiϕx − 1∣∣∣+ (ϕ+ 1)s − 1

]·

√√√√ N∑−N

|an · cn · (2n)s|2

are valid at s = 1, 2, 3, ...The proof of Lemmas 1.1 and 1.2 is given in [8].Using lemmas 1.1 and 1.2, we obtainLemma 1.3. Let an be a finite system of complex numbers. Then the

following inequality ∥∥∥∥∥N∑−N

an(yn(x)− zn(x))

∥∥∥∥∥W s

2 (0,π)

≤

≤

√θ2 + 2

( θ√2

+ (ϕ+ 1)s − 1)2 · σ(s) ·

√√√√ N∑−N

|an|2

is valid where σ(0) =1√2, σ(s) = 1 at s > 0.

Lemma 1.4. Let α 6= 0, β 6= 0, |α| 6= |β| be real numbers, and

ρ =

√θ2 + 2

(θ√2

+ (ϕ+ 1)s − 1)2

· σ(s) < 1

where σ(0) =1√2, σ(s) = 1 at s > 0, θ =

√2 · max

x∈[0,π]

∣∣eiϕx − 1∣∣, λn =

2n + εn · ϕ, ϕ =1π


, εn = ε−n = ±1 at n ∈ Z. Then the

system of eigenfunctions

yn(x) =

√2π· β cosλnx+ εn · sign(β2 − α2) · α sinλnx√

α2 + β2 ·√

1 + (2n)2s, n ∈ Z,

of the spectral problem (1.6)-(1.7) forms the Riesz basis in the spaceW s

2 (0, π).Proof of Lemma 1.4. The system of vectors

zn(x) =

√2π· β cos 2nx+ εn · sign(β2 − α2) · α sin 2nx√

α2 + β2 ·√

1 + (2n)2s, n ∈ Z


forms the complete orthonormal system in the Hilbert space W s2 (0, π), and

the system of vectors

yn(x) =


α2 + β2 ·√

1 + (2n)2s, n ∈ Z

by virtue of Lemma 1.3 satisfies the conditions of the theorem by R. Paleyand N. Wiener (see. [5], p. 224). This theorem implies that the system ofvectors yn(x)n∈Z forms the Riesz basis in the space W s

2 (0, π).Lemma 1.5. The operator Ly = −y′′ with the domain

D(L) = y(x) : y(x) ∈ C2(0, π) ∩ C1[0, π], y′′ ∈ L2(0, π),

αy(0) + βy(π) = 0, βy′(0) + αy′(0) = 0

is a symmetric operator in the class L2(0, π).The proof of Lemma 1.5 is given in [8].Theorem 1.1. Let α 6= 0, β 6= 0, |α| 6= |β| be real number, and

ρ =

√θ2 + 2

( θ√2

+ (ϕ+ 1)s − 1)2 · σ(s) < 1

where σ(0) =1√2, σ(s) = 1 at s > 0, θ =

√2 · max

x∈[0,π]

∣∣eiϕx − 1∣∣, λn =

2n + εn · ϕ, ϕ =1π


, εn = ε−n = ±1 at n ∈ Z. Then the

system of eigenfunctions

yn(x) =


α2 + β2 ·√

1+ | λn |2s, n ∈ Z,

of the spectral problem (1.6)-(1.7) form the complete orthonormal system inthe Sobolev classes W s

2 (0, π).Proof of Theorem 1.1. Symmetry of the operator L implies that

eigenfunctions yn(x)n∈Z of the operator L corresponding to the differenteigenvalues are orthogonal in classes L2(0, π).

The system of functions Dαyn(x)n∈Z is also the system ofeigenfunctions of a similar operator corresponding to different eigenvalues,what implies that functions of the system Dαyn(x)n∈Z are orthogonal inclasses L2(0, π).


As a result, we get that the system of eigenfunctions yn(x)n∈Z ofthe operator L corresponding to different eigenvalues are orthogonal in theSobolev classes W s

2 (0, π).It is known, if a sequence of vectors ψn(x)n∈Z forms the Riesz basis

in a Hilbert space H, then the system of vectorsψn(x)

n∈Z

,(ψn(x) =

ψn(x)‖ψn(x)‖

, n ∈ Z)

also forms the Riesz basis in H (see [7], p.374).

By virtue of Lemma 1.4, the system of eigenvectors yn(x)n∈Z formsthe Riesz basis in the space W s

2 (0, π). Orthogonality of this system impliesthat yn(x)n∈Z is a complete orthonormal system in the Sobolev classesW s

2 (0, π). Theorem 1 is proved.Theorem 1.1 and the Sobolev embedding theorem imply the following

corollaries.Corollary 1.1. Let α 6= 0, β 6= 0, |α| 6= |β| be real numbers, and

ρ =

√θ2 + 2

(θ√2

+ ϕ

)2

< 1

where θ =√

2 · maxx∈[0,π]

∣∣eiϕx − 1∣∣, λn = 2n + εn · ϕ, ϕ =

1π


,

εn = ε−n = ±1 at n ∈ Z. Then the Fourier series for the function f(x) ∈W 1

2 (0, π) ∩ C[0, π] in orthonormal eigenfunctions

yn(x) =


α2 + β2 ·√

1+ | λn |2, n ∈ Z

of the spectral problem (1.6)-(1.7) converges uniformly on the segment [0, π]to the function f(x).

Corollary 1.2. Let α 6= 0, β 6= 0, |α| 6= |β| be real numbers, and

ρ =

√θ2 + 2

(θ√2

+ (ϕ+ 1)s − 1)2

< 1

where s > k, θ =√

2· maxx∈[0,π]

∣∣eiϕx−1∣∣, λn = 2n+εn·ϕ, ϕ =

1π


,

εn = ε−n = ±1 at n ∈ Z. Then the Fourier series for the function f(x) ∈W s

2 (0, π) ∩ Ck[0, π] in orthonormal eigenfunctions

yn(x) =


α2 + β2 ·√

1+ | λn |2, n ∈ Z,


of the spectral problem (1.6)-(1.7) converges in the norm of the spaceCk[0, π] to the function f(x).

The scalar product in the space W s1,s22 ((0, π) × (0, π)) is introduced in

a following way:

(f(x, y), g(x, y))W s1,s22 ((0,π)×(0,π)) = (f(x, y), g(x, y))L2((0,π)×(0,π))+

+(Ds1x f(x, y), Ds1

x g(x, y))L2((0,π)×(0,π))+

+(Ds2y f(x, y), Ds2

y g(x, y))L2((0,π)×(0,π))+

+(Ds1,s2x,y f(x, y), Ds1,s2

x,y g(x, y))L2((0,π)×(0,π)).

Respectively, the norm in this space is introduced as follows:∥∥∥f(x, y)∥∥∥2

Ws1,s22 ((0,π)×(0,π))

=∥∥∥f(x, y)

∥∥∥2

L2((0,π)×(0,π))+

+∥∥∥Ds1

x f(x, y)∥∥∥2

L2((0,π)×(0,π))+

+∥∥∥Ds2

y f(x, y)∥∥∥2

L2((0,π)×(0,π))+∥∥∥Ds1,s2

x,y f(x, y)∥∥∥2

L2((0,π)×(0,π)).

Lemma 1.6. If ϕm(x) and ψn(y) be complete orthonormal systemsin W s1

2 (0, π) and W s22 (0, π), respectively, then the system of all products

fmn(x, y) = ϕm(x) · ψn(y)

is a complete orthonormal system in W s1,s22 ((0, π)× (0, π)).

Proof of Lemma 1.6. By virtue of the Fubini theorem,∥∥∥fmn(x, y)∥∥∥2

Ws1,s22 ((0,π)×(0,π))

=∥∥∥ϕm(x)

∥∥∥2

L2(0,π)·∥∥∥ψn(y)∥∥∥2

L2(0,π)+

+∥∥∥Ds1

x ϕm(x)∥∥∥2

L2(0,π)·∥∥∥ψn(y)∥∥∥2

L2(0,π)+∥∥∥ϕm(x)

∥∥∥2

L2(0,π)·∥∥∥Ds2

y ψn(y)∥∥∥2

L2(0,π)+

+∥∥∥Ds1

x ϕm(x)∥∥∥2

L2(0,π)·∥∥∥Ds2

y ψn(y)∥∥∥2

L2(0,π)=

=(∥∥∥ϕm(x)

∥∥∥2

L2(0,π)+∥∥∥Ds1

x ϕm(x)∥∥∥2

L2(0,π)

)·∥∥∥ψn(y)∥∥∥2

L2(0,π)+

+(∥∥∥ϕm(x)

∥∥∥2

L2(0,π)+∥∥∥Ds1

x ϕm(x)∥∥∥2

L2(0,π)

)·∥∥∥Ds2

y ψn(y)∥∥∥2

L2(0,π)=


=(∥∥∥ϕm(x)

∥∥∥2

L2(0,π)+∥∥∥Ds1

x ϕm(x)∥∥∥2

L2(0,π)

)·

·(∥∥∥ψn(y)∥∥∥2

L2(0,π)+∥∥∥Ds2

y ψn(y)∥∥∥2

L2(0,π)

)= 1.

If m 6= m1 or n 6= n1, by the same theorem

(fmn(x, y), fm1n1(x, y))W s1,s22 ((0,π)×(0,π)) =

= (fmn(x, y), fm1n1(x, y))L2((0,π)×(0,π))+

+(Ds1x fmn(x, y), D

s1x fm1n1(x, y))L2((0,π)×(0,π))+

+(Ds2y fmn(x, y), D

s2y fm1n1(x, y))L2((0,π)×(0,π))+

+(Ds1,s2x,y fmn(x, y), Ds1,s2

x,y fm1n1(x, y))L2((0,π)×(0,π)) =

= (ϕm(x), ϕm1(x))L2(0,π) · (ψn(y), ψn1(y))L2(0,π)+

+(Ds1x ϕm(x), Ds1

x ϕm1(x))L2(0,π) · (ψn(y), ψn1(y))L2(0,π)+

+(ϕm(x), ϕm1(x))L2(0,π) · (Ds2y ψn(y), D

s2y ψn1(y))L2(0,π)+

+(Ds1x ϕm(x), Ds1

x ϕm1(x))L2(0,π) · (Ds2y ψn(y), D

s2y ψn1(y))L2(0,π) =

= ((ϕm(x), ϕm1(x))L2(0,π)+

+(Ds1x ϕm(x), Ds1

x ϕm1(x))L2(0,π)) · (ψn(y), ψn1(y))L2(0,π)+

+((ϕm(x), ϕm1(x))L2(0,π)+

+(Ds1x ϕm(x), Ds1

x ϕm1(x))L2(0,π)) · (Ds2y ψn(y), D

s2y ψn1(y))L2(0,π) =

= ((ϕm(x), ϕm1(x))L2(0,π)+

+(Ds1x ϕm(x), Ds1

x ϕm1(x))L2(0,π))·

·((ψn(y), ψn1(y))L2(0,π) + (Ds2y ψn(y), D

s2y ψn1(y))L2(0,π)) = 0

since the scalar product (fmn(x, y), fm1n1(x, y))W s1,s22 ((0,π)×(0,π)) of two

variables exist on Π = (0, π)× (0, π).Let’s prove completeness of the system fmn(x, y). Assume that there

exists a function f(x, y) in W s1,s22 ((0, π)× (0, π)) which is orthogonal to all

functions fmn(x, y). Set Fm(y) = (f(x, y), ϕm(x))W s12 (0,π). It is easy to see,

that the function Fm(y) belongs to the class W s22 (0, π). That’s why for any

n, again applying the Fubini theorem, we obtain

(Fm(y), ψn(y))W s22 (0,π) = (f(x, y), fmn(x, y))W s1,s2

2 ((0,π)×(0,π)) = 0.


By completeness of the system ψn(y), for almost all y

Fm(y) = 0.

But then for almost every y, the equalities

(f(x, y), ϕm(x))W s12 (0,π) = 0

hold for all m. Completeness of the system ϕm(x) implies that for almost ally the set of those x, for which f(x, y) 6= 0, has the measure zero. By virtueof the Fubini theorem, it means that on Π = (0, π) × (0, π), the functionf(x, y) is zero almost everywhere. Lemma 1.6 is proved.

The scalar product in the space W s1,s2,...,sN2 (Π) is introduced in a

following way:

(f(x), g(x))W s1,s2,...,sN2 (Π) = (f(x), g(x))L2(Π)+

+N∑j1=1

(Dsj1xj1f(x), Dsj1

xj1g(x))L2(Π)+

+∑

1≤j1<j2≤N

(Dsj1xj1Dsj2xj2f(x), Dsj1

xj1Dsj2xj2g(x))L2(Π) + · · ·+

+∑

1≤j1<j2<···<jN≤N

(Dsj1xj1Dsj2xj2

. . . DsjNxjN

f(x), Dsj1xj1Dsj2xj2

. . . DsjNxjN

g(x))L2(Π).

Respectively, the norm in this space is introduced as follows:

∥∥∥f(x)∥∥∥2

Ws1,s2,...,sN2 (Π)

=∥∥∥f(x)

∥∥∥2

L2(Π)+

N∑j1=1

∥∥∥Dsj1xj1f(x)

∥∥∥2

L2(Π)+

+∑

1≤j1<j2≤N

∥∥∥Dsj1xj1Dsj2xj2f(x)

∥∥∥2

L2(Π)+ · · ·+

+∑

1≤j1<j2<···<jN≤N

∥∥∥Dsj1xj1Dsj2xj2

. . . DsjNxjN

f(x)∥∥∥2

L2(Π).

Using the method of mathematical induction and Lemma 1.6, we obtainthe following


Lemma 1.6. If ϕm1(x1), . . . , ϕmN (xN ) are complete orthonormalsystems in the spaces W s1

2 (0, π), . . . , W sN2 (0, π), respectively, then the

system of all products

fm(x) = fm1···N (x1, . . . , xN ) = ϕm1(x1) · · · · · ϕmN (xN )

is a complete orthonormal system in W s1,s2,...,sN2 (Π).

Let’s apply Lemma 1.7 to our orthonormal systems. In the spaceW s1,s2,...,sN

2 (Π) of functions of N variables f(x) = f(x1, . . . , xN ) allproducts

vm1···N (x1, . . . , xN ) = ym1(x1) · · · · · ymN (xN )

form the complete orthonormal system. Here

ymj (xj) =

√2π·βj cosλmjxj + εmj · sign(β2

j − α2j ) · αj sinλmjxj√

α2j + β2

j ·√

1+ | λmj |2sj, mj ∈ Z

at 1 ≤ j ≤ p,

ymj (xj) =1√π

1√1+ | 2mj |2sj

exp(i2mjxj), mj ∈ Z

at p+ 1 ≤ j ≤ q,

ymj (xj) =

√2π

1√1+ | mj |2sj

sin(mjxj), mj ∈ N

at q + 1 ≤ j ≤ N.Thus, the following statement is valid.Theorem 1.2. Let αj 6= 0, βj 6= 0, |αj | 6= |βj | be real numbers at every

1 ≤ j ≤ p, and

ρ = max1≤j≤p

√θ2j + 2

( θj√2

+ (ϕj + 1)sj − 1)2 · σ(sj) < 1

where σ(0) =1√2, σ(sj) = 1, at sj > 0, θj =

√2 · max

x∈[0,π]

∣∣eiϕjx − 1∣∣,

λmj = 2mj+εmj ·ϕj , ϕj =1π

arccos−2αjβjα2j + β2

j

, εmj = ε−mj = ±1 at mj ∈ Z.

Then the system of eigenfunctions

vm1···N (x1, . . . , xN )(m1,...,mp)∈Zp,(mp+1,...,mq)∈Zq−p,(mq+1,...,mN )∈NN−q =


=

p∏j=1

√2π·βj cosλmjxj + ε

mj· sign(β2


α2j + β2

j ·√

1+ | λmj |2sj

(m1,...,mp)∈Zp×

q∏

j=p+1

1√π

1√1+ | 2mj |2sj

exp(i2mjxj)

(mp+1,...,mq)∈Zq−p

×

×

N∏

j=q+1

√2π

1√1+ | mj |2sj

sin(mjxj)

(mq+1,...,mN )∈NN−q

of the spectral problem (1.4)-(1.5) forms the complete orthonormal systemin the Sobolev classes W s1,s2,...,sN

2 (Π).Corollary 1.3. Let αj 6= 0, βj 6= 0, |αj | 6= |βj | be real numbers at every

1 ≤ j ≤ p, and

ρ = max1≤j≤p

√θ2j + 2

(θj√2

+ (ϕj + 1)sj − 1)2

· σ(sj) < 1

where σ(0) =1√2, σ(sj) = 1 at sj > 0, θj =

√2 · max

x∈[0,π]

∣∣eiϕjx − 1∣∣, λmj =

2mj + εmj · ϕj , ϕj =1π


j

, εmj = ε−mj = ±1 at mj ∈ Z,

sj > k +N

2, k ≥ 0, k ∈ Z. Then the Fourier series for the function f(x) ∈

W s1,s2,...,sN2 (Π) ∩ Ck(Π) in orthonormal eigenfunctions

vm1···N (x1, . . . , xN )(m1,...,mp)∈Zp,(mp+1,...,mq)∈Zq−p,(mq+1,...,mN )∈NN−q =

=

p∏j=1

√2π·βj cosλmjxj + εmj · sign(β2


α2j + β2

j ·√

1+ | λmj |2sj

(m1,...,mp)∈Zp×

q∏

j=p+1

1√π

1√1+ | 2mj |2sj

exp(i2mjxj)

(mp+1,...,mq)∈Zq−p

×

×

N∏

j=q+1

√2π

1√1+ | mj |2sj

sin(mjxj)

(mq+1,...,mN )∈NN−q


of the spectral problem (1.4)-(1.5) converges in the norm of the space Ck(Π)to the function f(x).

The proof of Corollary 1.3 is carried out using Theorem 1.2 and theSobolev embedding theorem.

Is is true the followingTheorem 1.3. Let αj 6= 0, βj 6= 0, |αj | 6= |βj | be real numbers at every

1 ≤ j ≤ p, and

ρ = max1≤j≤p

√θ2j + 2

( θj√2

+ (ϕj + 1)sj − 1)2 · σ(sj) < 1

where σ(0) =1√2, σ(sj) = 1 at sj > 0, θj =

√2 · max

x∈[0,π]

∣∣eiϕjx − 1∣∣, λmj =

2mj + εmj · ϕj , ϕj =1π


j

, εmj = ε−mj = ±1 at mj ∈ Z,

sj > k + N2 , k ≥ 0, k ∈ Z and ϕj(x) ∈ W

s1+j−N2 ,s2+j−

N2 ,...,sN+j−N

22 (Π),

f(x, t) ∈ Ws1,s2,...,sN ,sN+12 (Π × (0,+∞)). Then the solution of the problem

(11.)-(1.2)-(1.3) exists, it is unique and is represented in the form of theseries

u(x, t) =∞∑

m1=−∞· · ·

∞∑mN=−∞

n∑j=1

ϕj,(m1···N )tα−jEα,α−j+1(−µm1···N · tα)+

+

t∫0

(t− τ)α−1 · Eα,α[−µm1···N (t− τ)α]fm1···N (τ)dτ

·vm1...mN (x1, . . . , xN ),

(1.16)where coefficients are determined in a following way :

Eα,α−j+1(−µm1···N · tα) =∞∑i=0

(−µm1···N · tα)i

Γ(αi+ α− j + 1), (1.17)

Eα,α

(− µm1···N · (t− τ)α

)=

∞∑i=1

(−µm1···N )i−1 · (t− τ)α(i−1)

Γ(α · i), (1.18)

f(x, t) =∞∑

m1=−∞· · ·

∞∑mN=−∞

fm1···N (t) · vm1···N (x1, . . . , xN ), (1.19)


ϕj(x) =∞∑

m1=−∞· · ·

∞∑mN=−∞

ϕj,(m1···N ) · vm1···N (x1, . . . , xN ), j = 1, 2, . . . , n.

(1.20)

Proof of Theorem 1.3. S ince the system of eigenfunctions

vm1···N (x1, . . . , xN )(m1,...,mp)∈Zp,(mp+1,...,mq)∈Zq−p,(mq+1,...,mN )∈NN−q

of the spectral problem (1.4)-(1.5) forms the complete orthonormalsystem in the Sobolev classes W s1,s2,...,sN

2 (Π), any function from the classW s1,s2,...,sN

2 (Π) can be represented as a convergent Fourier series in thissystem. For any t > 0, expand the solution u(x, t) of the problem (1.1)-(1.2)-(1.3) into the Fourier series in eigenfunctions

vm1···N (x1, . . . , xN )(m1,...,mp)∈Zp,(mp+1,...,mq)∈Zq−p,(mq+1,...,mN )∈NN−q

of the spectral problem (1.4)-(1.5):

u(x, t) =∞∑

m1=−∞· · ·

∞∑mN=−∞

Tm1···N (t) · vm1···N (x),

Tm1···N (t) = (u(x, t), vm1···N (x)). (1.21)

By virtue of (1.1)-(1.2), unknown functions Tm(t) must satisfy the equation

Dα0tTm1···N (t) + λm1···NTm1···N (t) = fm1···N (t), n− 1 < α ≤ n, n ∈ N

(1.22)with initial conditions

limt→0

Dα−k0t Tm1···N (t) = ϕk,m1···N , k = 1, 2, . . . , n, m ∈ Z. (1.23)

The solution of the Cauchy problem (1.22)-(1.23) has the form

Tm1···N (t) =n∑j=1

ϕj,(m1···N )tα−jEα,α−j+1(−µm1···N · tα)+

+

t∫0

(t− τ)α−1 · Eα,α[−µm1···N (t− τ)α]fm1···N (τ)dτ (1.24)


where coefficients are determined as follows:

Eα,α−j+1(−µm1···N · tα) =∞∑i=0

(−µm1···N · tα)i

Γ(αi+ α− j + 1), (1.25)

Eα,α

(− µm1···N · (t− τ)α

)=

∞∑i=1

(−µm1···N )i−1 · (t− τ)α(i−1)

Γ(α · i), (1.26)

f(x, t) =∞∑

m1=−∞· · ·

∞∑mN=−∞

fm1···N (t) · vm1···N (x1, . . . , xN ), (1.27)

ϕj(x) =∞∑

m1=−∞· · ·

∞∑mN=−∞

ϕj,(m1···N ) · vm1···N (x1, . . . , xN ), j = 1, 2, . . . , n.

(1.28)After substituting (1.24) into (1.21), we obtain the unique solution of theproblem (1.1)-(1.2)-(1.3) in the form of the series (1.16). Theorem 1.3 isproved.

References

1. Naimark M.A., Linear Differential Operators, “Nauka”, Moscow. 1969.p.528.

2. Levitan B.M., Sargsyan I.S., Introduction to Spectral Theory:Selfadjoint Ordinary Differential Operators, “Nauka”, Moscow. 1970.p.672; English transl., Transl. Math. Monographs, vol. 39, Amer.Math. Soc., Providence, RI, 1975.

3. Levitan B.M., Sargsyan I.S., Sturm-Liouville and Dirac Operators,“Nauka”, Moscow, 1988. p.432.

4. Kostychenko A.G., Sargsyan I.S., Distribution of Eigenvalues:Selfadjoint Ordinary Differential Operators, “Nauka”, Moscow, 1979.p.400.

5. Riesz F., Szokefalvi-Nagy B., Functional Analysis, New York F. UngarPub. Co., 1955. p.592.

6. Sadovnichiy V.A., Theory of Operators, MSU publishing house,Moscow, 1986. p.368.


7. Gokhberg I.Ts. and Krein M.G., Introduction to the Theory of LinearNonselfadjoint Operators in Hilbert Space, “Nauka”, Moscow, 1965. p.448; English transl., Amer. Math. Soc., Providence, R.I., 1969.

8. Kasimov Sh.G., Ataev Sh.K., On completeness of the systemof orthonormal vectors of a generalized spectral problem, UzbekMathematical Journal, 2009, No,2 , p.101-111.

9. Kasimov Sh.G., Ataev Sh.K., Madraximov U.S., On solvability ofthe mixed problem for a partial equation of a fractional order withSturm-Liouville operators and nonlocal boundary conditions, UzbekMathematical Journal, 2016, No,2 , p.158-169.

Kasimov Sh.G.National University of Uzbekistan, University street, 4, 100174,Tashkent, Uzbekistan, e-mail: [email protected] Sh.K.National University of Uzbekistan, University street, 4, 100174,Tashkent, Uzbekistan, e-mail: otaev−[email protected]

90 Khalkulova Kh.A., Abdurasulov K.K.


Solvable Lie algebras with maximal dimension ofcomplementary space to nilradicalKhalkulova Kh.A., Abdurasulov K.K.

Abstract. This work is devoted to the study of solvable Lie algebras for whichthe dimension of the complementary space is equal to the number of generatorsof the nilradical. The description of such classes of solvable Lie algebras simplifiesthe process of classification of solvable algebras with a given nilradical.

Keywords: Lie algebra, solvable algebra, nilradical, complementary spaceMathematics Subject Classification (2010): 17A32, 17A36,17B30,17B56

1 Introduction

The theory of Lie algebras is a classically well known object which is activelystudied in the last century. The study of Lie algebras does not lose itsrelevance to the present day. From the classical theory of Lie algebras it isknown that any finite dimensional complex Lie algebra can be decomposedinto the semidirect sum of its solvable radical and a semisimple Lie algebra.The semisimple part can be described by simple Lie ideals. Therefore, themain focus in the investigation of finite dimensional complex Lie algebrasis to study of solvable radical.

A.I. Malcev proved that a solvable Lie algebra is uniquely determinedby its nilradical [1]. Mubarakzjanov proposed the method, which usedthe property that the dimension of the complementary vector space doesnot exceed the number of nil-independent derivations and the number ofgenerators of the nilradical [2].

Using this method, the classification of solvable Lie algebras with theAbelian, Heisenberg, filiform, quasi-filiform nilradicals are obtained.

This paper is devoted to the study of solvable Lie algebras for which thedimension of the complementary space is equal to the number of generatorsof the nilradical. The description of such classes of solvable Lie algebrassimplifies the process of classification of solvable algebras with a givennilradical.

Solvable Lie algebras with maximal ... 91

2 PreliminariesDefinition 2.1. A vector space with bilinear bracket (G, [−,−]) over a fieldF is called a Lie algebra if for any x, y, z ∈ G the following conditions arehold:

[x, y] = −[y, x], – antisymmetry,

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, – Jacobi identity.

For an arbitrary Lie algebra L, we define the following series:

L[1] = L, L[n+1] = [L[n], L[n]]; L1 = L,Ln+1 = [Ln, L], n ≥ slant1.

Definition 2.2. Algebra L is said to be nilpotent (respectively, solvable),if there exists n ∈ N (respectively,m ∈ N) such that L[n] = 0 (respectively,Lm = 0). The minimal number n(respectively,m) such that the property isholds said to be the index of nilpotency (respectively, index of solvability)of the algebra L

The maximal nilpotent ideal of an algebra is called a nilradical.

3 Main resultsLet R be a solvable Lie algebra with nilradical N (with the index ofnilpotency τ), and let Q be a complementary space of the nilradical.Then we have R = N ⊕ Q. Let x1, x2, . . . , xk be a basis of Q andE = e1, e2, . . . , es ⊂ N be a set of generators of the nilradical N .

Below we introduce some denotations which will be used later on.The space M of all right normed words of length m of the alphabet E

can be decomposed into three subspace M0,M1,M2, where M0 is a subspaceof M with no entries of e1, M1 is the subspace of M with only one entry ofe1 and M2 is the subspace of M with more than one entry of e1.

Moreover, we have

M0 =k⋃t=2

M0,t, M1 =k⋃t=2

M1,t,

where M0,t is a subspace of M0 span of the words with minimal subindexof e2, e3, . . . , es equal to t and M1,t is a subspace of M1 span of the wordswith minimal subindex of e2, e3, . . . , es equal to t.


We set

ut,1, ut,2, . . . , ut,st, vt,1, vt,2, . . . , vt,kt, w1, w2, . . . , wp

the bases of M0,t,M1,t,M2, respectively.From Jacobi identity we conclude that any words decomposed into a

linear combination of rightnormed words of the same length. Therefore,without loss of generality we can assume that basis elements ut,i, vt,j , wlcan be chosen as rightnormed words of alphabet E.

It is obvious that the space Nm forms an ideal of R for any m. In thefollowing theorem we shall describe some products in R/Nm.

Theorem 3.1. Let R = N ⊕Q be a solvable Lie algebra and k = s. ThenR admits a basis e1, e2, . . . , en, x1, x2, . . . , xk such that the multiplicationof N ⊕Q has the following form:

[ei, xi] = ei, 1 ≤ i ≤ k,

[ei, xj ] = 0, 1 ≤ i 6= j ≤ k,

[xi, xj ] = 0, 1 ≤ i, j ≤ k.

(3.1)

Proof. First we note that R/Nm = (N ⊕ Q)/Nm = N/Nm ⊕ Q. Thismeans that the quotient algebra R/Nm has nilradical N/Nm.

In order to prove the theorem we use induction on m (2 ≤ m ≤ τ + 1).It suffices to the products in R/Nm. Since, if m = tau + 1, then R =R/N tau+1.

For m = 2, the nilradical N/N2 is abelian and the products (1) followfrom [3]. Let us assume that the products (1) are true for all m. In orderto prove (2) for all m in the quotient algebra R/Nm+1 we need to considerthe products:

[ei, xi], [ei, xj ], [xi, xj ], 1 ≤ i 6= j ≤ k.

For convenience we shall use congruences without indicating moduloNm+1.

By induction we have:

[e1, x1] ≡ e1 +k∑j=2

sj∑l=1

α1j,luj,l +

k∑j=2

kj∑l=1

β1j,lvj,l +

p∑q=1

γ1qwq,

[e1, xi] ≡k∑j=2

sj∑l=1

αij,luj,l +k∑j=2

kj∑l=1

βij,lvj,l +p∑q=1

γiqwq, 2 ≤ i ≤ k.


Note that [wq, x1] ≡ lqwq with lq ≥ 2. Indeed, applying the Jacobiidentity [[x, y], z] = [x, [y, z]] + [[x, z], y] and induction hypothesis we derive

[wq, x1] = [[[...[ei1 , ei2 ], ei3 ], ...], eim ], x1] ≡ lqwq

where lq = αim + αim−1 + ...+ αi2 + αi1 with αij =

1, if ij = 1,

0, if ij 6= 1.

Since the entries of e1 in wq are more than one times we conclude thatlq ≥ 2 .

Similarly, we derive [vi,l, xi] ≡ ri,lvi,l, ri,l ≥ 1.Taking the following change:

e′1 = e1 +k∑j=2

sj∑l=1

αj,luj,l −k∑j=2

kj∑l=1

βjj,lrj,l

vj,l +p∑q=1

γq1− lq

wq,

one can assume

[e1, x1] ≡ e1 +k∑j=2

kj∑l=1

βj,lvj,l

and

[e1, xi] ≡k∑j=2

sj∑l=1

αij,luj,l +k∑

j=2, j 6=i

kj∑l=1

βij,lvj,l +p∑q=1

γiqwq, 2 ≤ i ≤ k.

Consider the Jacobi identity

0 ≡ [e1, [x1, xi]] = [[e1, x1], xi]− [[e1, xi], x1] ≡

≡k∑j=2

sj∑l=1

αij,luj,l +ki∑l=1

ri,lβi,lvi,l +i−1∑j=2

kj∑l=1

β′j,lvj,l +p∑q=1

(1− lq)γiqwq,

where 2 ≤ i ≤ k.Comparing the coefficients at the basis elements we deduce

αij,l = 0, 1 ≤ l ≤ sj , βi,l = 0, 1 ≤ l ≤ ki, γiq = 0, 1 ≤ q ≤ p, 2 ≤ i, j ≤ k.

Therefore, we obtain

[e1, x1] ≡ e1, [e1, xi] ≡k∑

j=2, j 6=i

kj∑l=1

βij,lvj,l.


Now for i and q (2 ≤ i < q ≤ k) we consider Jacoby identity

0 ≡ [e1, [xi, xq]] = [[e1, xi], xq]− [[e1, xq], xi] ≡

≡kq∑l=1

rq,lβiq,lvq,l +

q−1∑j=2

kj∑l=1

βij,l′vj,l −

ki∑l=1

ri,lβqi,lvi,l −

i−1∑j=2

kj∑l=1

βqj,l′vj,l,

Comparing the coefficients of the basis elements we derive

βiq,l = 0, 2 ≤ i < q ≤ k, 1 ≤ l ≤ kq.

Thus, we have

[e1, x2] ≡ 0, [e1, xi] ≡i−1∑j=2

kj∑l=1

βij,lvj,l, 3 ≤ i ≤ k. (3.2)

By induction we shall prove that

[e1, xh] ≡ 0, 2 ≤ h ≤ i− 1, [e1, xi] ≡i−1∑j=h

kj∑l=1

βij,lvj,l, 3 ≤ i ≤ k. (3.3)

Suppose that the equality 3.3 is true, then for h = 2 we obtain theequality (3.2).

Considering following chain of equalities

0 ≡ [e1, [xi, xh]] = [[e1, xi], xh]− [[e1, xh], xi] ≡kh∑l=1

rh,lβih,lvh,l

we establish that βih,l = 0, 1 ≤ l ≤ kh.Therefore, we get

[e1, x1] ≡ e1, [e1, xi] ≡ 0.

Applying the previous argument for all values of i to the products [ei, xj ]and taking into account that the used changes of basis do not affect theproducts [et, xj ] for 1 ≤ t ≤ i− 1, we deduce the products

[ei, xi] ≡ ei, 1 ≤ i ≤ k,

[ei, xj ] ≡ 0, 1 ≤ i 6= j ≤ k.


In order to prove [xi, xj ] ≡ 0 for 1 ≤ i, j ≤ k we need to consider thesubspaces Mt, 1 ≤ t ≤ k, where Mt is a subspace of M span of the wordswith minimal subindex of e1, e2, . . . , ek equal to t.

As we mentioned above without loss of generality one can assume thatbasis elements vt,j , 1 ≤ j ≤ sj of Mt can be chosen as rightnormed wordsof alphabet E.

Let us introduce notation

[x1, xi] ≡s1∑q=1

αq1,iv1,q +si∑q=1

βq1,ivi,q +k∑

l=2, l 6=i

sl∑q=1

γl,q1,ivl,q, 2 ≤ i ≤ k,

[xi, xj ] ≡si∑q=1

αqi,jvi,q +sj∑q=1

βqi,jvj,q +k∑

l=1, l 6=i,j

sl∑q=1

γl,qi,jvl,q, 2 ≤ i, j ≤ k.

Clearly, [vt,i, xt] = pt,ivt,i with pt,i ≥ 1.

Taking the change x′1 = x1 −k∑t=2

st∑l=1

βl1,tpt,l

vt,l, for 2 ≤ i ≤ k one canassume,

[x′1, xi] ≡s1∑q=1

αq1,iv1,q +k∑

t=2,t6=i

st∑q=1

γt,q1,ivt,q.

For a pair i, j such that 2 ≤ i < j ≤ k we have

[x1, [xi, xj ]] = [[x1, xi], xj ]− [[x1, xj ], xi] ≡

≡s1∑q=1

αq1,i′v1,q +

i−1∑t=2

st∑q=1

γt,q1,i

′vt,q +

j−1∑t=i+1

st∑q=1

γt,q1,i [vt,q, xj ]+

+sj∑q=1

pj,qγj,q1,ivj,q −

i−1∑t=1

st∑q=1

γt,q1,j

′vt,q −

si∑q=1

pi,qγi,q1,jvi,q.

On the other hand,

[x1, [xi, xj ]] ≡ [x1,

si∑q=1

αqi,jvi,q+sj∑q=1

βqi,jvj,q+k∑

t=1, l 6=i,j

st∑q=1

γt,qi,j vt,q] ≡s1∑q=1

µqv1,q.

Consequently,

γj,q1,i = 0, 1 ≤ q ≤ sj , γi,q1,j = 0, 1 ≤ q ≤ si,


for 2 ≤ i < j ≤ k, which means

[x1, xi] ≡s1∑q=1

αq1,iv1,q, 2 ≤ i ≤ k.

Taking the change of basis elements x′i = xi+s1∑q=1

αq1,iri,q

v1,q, one can assume

[x1, xi] ≡ 0. Summarizing already obtained products, we have[ei, xi] = ei, 1 ≤ i ≤ k,

[ei, xj ] = 0, 1 ≤ i 6= j ≤ k,

[xi, xj ] = 0, 1 ≤ i 6= j ≤ k.

Theorem 3.2. Let R = N ⊕Q be a solvable Lie algebra such that dimQ =dimN/N2 = k. Then R admits a basis e1, e2, . . . , en, x1, x2, . . . , xk suchthat the table of multiplication in R has the following form:

[ei, ej ] =n∑

t=k+1

γti,jet, 1 ≤ i, j ≤ n,

[ei, xi] = ei, 1 ≤ i ≤ k,

[ei, xj ] = αi,jei, k + 1 ≤ i ≤ n, 1 ≤ j ≤ k,

where αi,j is the number of entries of a generator basis element ej involvedin forming non generator basis element ei.

Proof From Theorem 1 we have the existence a basise1, e2, . . . , en, x1, x2, . . . , xk of R with the products of N ⊕ Q hasthe form (1).

It should be noted that in the proof of Theorem 1, we used basis changesonly for generator elements of N . Therefore, without loss of generality, onecan assume that non generator basis elements of N are the rightnormedwords of the alphabet E.

Similarly to the proof of Theorem 1, we have

[ei, xj ] = [[[· · · [ei1 , ei2 ], ei3 ], . . . ], eit ], xj ] = αi,jei, k + 1 ≤ i ≤ n, 2 ≤ t ≤ τ,

where αi,j = αi1,j + αi2,j + ...+ αit−1,j + αit,j with αil,j =

1, if il = j,

0, if il 6= j,

for 1 ≤ l ≤ t.


Therefore, [ei, xj ] = αi,jei, k + 1 ≤ i ≤ n, 1 ≤ j ≤ k.The advantage of this theorem is that for any solvable algebras with the

condition k = s, we have the description of such solvable Lie algebras.Next, we give two examples of solvable Lie algebras obtained in [4],[5]

which can be obtained directly by using Theorem 3.2.

Example 3.3. In [4] the following result is proved: Precisely one class ofsolvable Lie algebras sn+2 of dimsn+2 = n+2 with nilradical nn,1 exists. Itis represented by a basis (e1, e2, . . . , en, f1, f2) and the Lie brackets involvingf1 and f2 are

[ek, en] = ek−1, 2 ≤ k ≤ n− 1,[f1, ek] = (n− 1− k)ek, 1 ≤ k ≤ n− 1,[f2, ek] = ek, 1 ≤ k ≤ n− 1,[f1, en] = en.

Using Theorem 1, we have products for the generator basis elements of thenilradical en−1, en.

[x, en−1] = −en−1, [y, en] = en.

Further, by Theorem 2, using the property of the nilradical, we write theremaining products:

[x, ek] = −ek, 1 ≤ k ≤ n− 1,[y, ek] = (k − n+ 1)ek, 1 ≤ k ≤ n− 1,[y, en] = en.

taking the change f ′2 = −x, f1 = −y, we obtain a solvable Lie algebraobtained in [4].

Example 3.4. In [5] the following solvable Lie algebra with three-dimensional complementary space:

[en, ei] = ei−1, 1 ≤ i ≤ n− 2, [h1, ei] = (n− i− 2)ei, 1 ≤ i ≤ n− 3,

[h2, ei] = ei, 1 ≤ i ≤ n− 2, [h3, en−1] = en−1, [h1, en] = en.

Using Theorem 2, we immediately obtain a similar result. Thus, if weknow the multiplications table of nilradical and the number generators equalto the dimension of the complementary space, then by Theorem 3.2 we havethe table of multiplications of solvable Lie algebras.


References

1. A.I. Malcev, Solvable Lie algebras, Amer. Math. Soc. Transl. 27 (1950).

2. G.M. Mubarakzjanov, On solvable Lie algebras, Izv. Vyssh. Uchebn. Zaved.Mat. 1(32) (1963) 114–123 (in Russian).

3. Adashev J.Q., Ladra M., Omirov B.A. Solvable Leibniz algebras withnaturally graded non-Lie p-filiform nilradicals, Communications in Algebra,2017 (10), p. 4329–4347.

4. L. Snobl, P. Winternitz, A class of solvable Lie algebras and their Casimirinvariants, Jour. of Physics A.-2005. (38). c. 2687–2700.

5. Yan Wang, Jie Lin, Shaoqiang Deng, Solvable Lie Algebras withQuasifiliform Nilradicals, Communications in Algebra, vol. 36 (11), 2008,p. 4052–4067.

Khalkulova Kh.A.Institute of Mathematics, Uzbekistan Akademy of Sciences,Mirzo Ulugbek street, 81, 100170, Tashkent, Uzbekistan, e-mail:[email protected] K.K.Institute of Mathematics, Uzbekistan Akademy of Sciences,Mirzo Ulugbek street, 81, 100170, Tashkent, Uzbekistan, e-mail:[email protected]

Local derivations of solvable Lie algebras ... 99


Local derivations of solvable Lie algebras with Abeliannilradical

Khudoyberdiyev A.Kh., Sultonova D.Y.

Abstract. In this work we investigate local derivations of solvable Liealgebras. We show that in the class of solvable Lie algebras there exist algebraswhich admit local derivations which are not ordinary derivation and algebras forwhich every local derivation is a derivation.

Keywords: Lie algebra, derivation, Local derivations, solvable, nilradicalMathematics Subject Classification (2010): 17A32

1 Introduction

The notions of local derivations were first introduced by R.V. Kadison in1990 [5]. The main problems concerning this notion are to find conditionsunder which local derivations become derivations and to present examplesof algebras with local derivations that are not derivations.

In previous years, the description of local derivations on commutativealgebras and algebras of measurable operators are obtained. Investigation oflocal derivations on Banach algebras, von Neumann algebras, the algebrasof measurable operators were initiated in papers [1], [2] and [3].

In [1] by Sh.A.Ayupov and K.K. Kudaybergenov local derivations onfinite-dimensional Lie algebras are studied. In particular, it is proved thatevery local derivation on a finite-dimensional semi-simple Lie algebra overan algebraically closed field of characteristic zero is a derivation. Moreover,examples of finite-dimensional nilpotent Lie algebra which admit localderivations which are not derivations are given.

In this paper we investigate local derivations of solvable Lie algebras.We show that in the class of solvable Lie algebras there exist algebras whichadmit local derivations which are not ordinary derivation and algebras forwhich every local derivation is a derivation.

More precisely, local derivations of solvable Lie algebras with abeliannilradical and one-dimensional complementary space are investigated. Anecessary and sufficient conditions under which local derivations solvableLie algebras with abelian nilradical and one-dimensional complementaryspace become derivations are found.

100 Khudoyberdiyev A.Kh., Sultonova D.Y.

2 PreliminariesDefinition 2.1. vector space L over a field F with a binary operation [−,−]is called a Lie algebra, if for any x, y, z ∈ L the following identities are hold:

[x, x] = 0 – anticommutative identity,[[x, y], z] + [[y, z], x] + [[z, x], y] = 0 – Jacobi identity

For an arbitrary Lie algebra L consider the following central lower andderived sequences

L1 = L, Lk+1 = [Lk, L], k ≥ 1,

L[1] = L, L[k+1] = [L[k], L[k]], k ≥ 1.

Definition 2.2. A Lie algebra L is called solvable (nilpotent) if there existsm ∈ N(s ∈ N), such that L[m] = 0 (Ls = 0). The maximal nilpotent idealof Lie algebra is called nilradical.

Definition 2.3. Linear operator d : L→ L is called a derivation, if

d([x, y]) = [d(x), y] + [x, d(y)], for any x, y ∈ L.

The set of all derivations of a Lie algebra L is a Lie algebra with respectto commutation operation and it is denoted by Der(L)

A linear operator ∆ is called a local derivation if for any x ∈ L thereexists a derivation dx : L→ L such that ∆(x) = dx(x).

3 Main resultsWe consider the following solvable Lie algebras [4]:

L1 : [e2, e1] = e2;L2(α) : [e2, e1] = e2, [e3, e1] = αe3;L3 : [e2, e1] = e2 + e3, [e3, e1] = e1;L4(α) : [e2, e3] = e4, [e2, e1] = e2,

[e3, e1] = αe3, [e4, e1] = (1 + α)e4;L5 : [e2, e3] = e4, [e2, e1] = e2 + e3,

[e3, e1] = e3, [e4, e1] = 2e3;L6(α, β) : [e2, e1] = e2, [e3, e1] = αe3, [e4, e1] = βe4;L7(α) : [e2, e1] = e2 + e3, [e3, e1] = e3, [e4, e1] = αe4;L8 : [e2, e1] = e2 + e3, [e3, e1] = e3 + e4, [e4, e1] = e4.


Proposition 3.1. Every local derivation of the algebrasL1, L2(α), L6(α, β) is a derivation, and the algebras L3, L4, L5, L7, L8

admits a local derivation which is not a derivation.

Proof. For the proof of the Proposition we describe the derivations ofthe algebras L1, L2(α), L3, L4, L5, L6(α, β), L7, L8. In particular, thematrix form of the derivations of the algebra L1 has the form:

Der(L1) =(

0 ξ10 ξ2

).

Let ∆ be a local derivation on L1 and let

∆(e1) = α1e1 + α2e2, ∆(e2) = α3e1 + α4e2.

On the other hand ∆(e1) = de1(e1) = ξ1e2, which implies α1 = 0. From∆(e2) = de2(e2) = ξ2e2, we have α3 = 0. Hence, we get ∆(e1) = α2e2,∆(e2) = α4e2, which derive ∆ ∈ Der(L1). Thus, any local derivation on L1

is a derivation.Similarly, it is proved that every local derivation of the algebras

L2(α), L6(α, β) is a derivation.Next we consider the algebra L3. It is not difficult to obtain that the

matrix form of the derivation of the algebra L3 has the form:

Der(L3) =

0 ξ1,2 ξ1,30 ξ2,2 ξ2,30 0 ξ2,2

.

Now let us define a linear operator ∆ on L3 by

∆(e1) = 0, ∆(e2) = 0, ∆(e3) = e3.

Evidently, ∆ is not a derivation. We show that this operator is a localderivation. For an arbitrary element x = η1e1 + η2e2 + η3e3 we have

∆(x) = ∆(η1e1 + η2e2 + η3e3) = η3e3.

Let dx be a derivation. Consider

dx(x) = dx(η1e1 + η2e2 + η3e3) = η1dx(e1) + η2dx(e2) + η3dx(e3) =

η1(ξ1,2e2 + ξ1,3e3) + η2(ξ2,2e2 + ξ2,3e3) + η3ξ2,2e3 =


(η1ξ1,2 + η2ξ2,2)e2 + (η1ξ1,3 + η2ξ2,3 + η3ξ2,2)e3.

It is not difficult to see that for any η1, η2, η3 exists ξ1,2, ξ1,3, ξ2,2, ξ2,3such that

η1ξ1,2 + η2ξ2,2 = 0, η1ξ1,3 + η2ξ2,3 + η3ξ2,2 = η3.

Thus, we proved that for any element x there exists a derivation dx forwhich ∆(x) = dx(x). Hence, ∆ is a local derivation.

Similarly, it is not difficult to show that there exists a local derivationof the algebras L4, L5, L7, L8 which is not a derivation.

Let L be a solvable Lie algebra with nilradical N and let dimN = n,dimL = n + 1. Then we take a basis x, e1, e2, . . . , en of L such thate1, e2, . . . , en basis of N.

It is known that the operator of right multiplication adx : N → N,adx(y) = [y, x],∀y ∈ N is a non nilpotent operator [6]. Moreover, ifnilradical N is an abelian ideal, then solvable algebra L is characterized bythe operator adx, i.е., two solvable algebras with abelian nilradical N areisomorphic if and only if the corresponding operators of right multiplicationhave the same Jordan forms.

For example, algebras L2(α) and L3 are three-dimensional solvable Liealgebras with two-dimensional abelian nilradical. Operators ade1 on algebrasL2(α) and L3 respectively have the form:(

1 00 α

),

(1 10 1

).

It is obvious that the operator ade1 on the algebra L2(α) is adiagonalized, and on the algebra L3 it is not a diagonalized.

According to Proposition 1 it is known that every local derivation onL2(α) ia a derivation, and algebra L3 has a local derivation which is not aderivation.

In the following theorem we give a necessary and sufficient conditionthat a solvable Lie algebra with abelian nilradical and one-dimensionalcomplementary space has local derivation which is not a derivation.

Theorem 3.2. Let L be a solvable Lie algebra with abelian nilradical N .Let dimL = dimN + 1 and x ∈ L \ N. Every local derivation on L is aderivation if and only if adx is a diagonalized operator.


Proof. Let x, e1, e2, . . . , en be a basis of L such that e1, e2, . . . en isa basis of N and let adx diagonalized operator on N . Then the Jordan formof the operator adx has the form:

adx =

λ1 0 0 . . . 00 λ2 0 . . . 00 0 λ3 . . . 0. . . . . . . . . . . . . . .0 0 0 . . . λn

Consequently,

[ei, x] = λiei, 1 ≤ i ≤ n.

Case 1. Let λi 6= λj for i 6= j. Let d ∈ Der(L), thend(x) = β1e1 + β2e2 + · · ·+ βnen,d(ei) = αi,1e1 + αi,2e2 + · · ·+ αi,nen, 1 ≤ i ≤ n.

From the property of derivation we have

d([ei, x]) = [d(ei), x] + [ei, d(x)] =

= [αi,1e1 + αi,2e2 + · · ·+ αi,nen, x] + [ei, β1e1 + β2e2 + · · ·+ βnen] =

= αi,1λ1e1 + αi,2λ2e2 + · · ·+ αi,nλnen.

On the other hand,

d([ei, x]) = λid(ei) = λi(αi,1e1 + αi,2e2 + · · ·+ αi,nen).

Comparing the coefficients at the basis elements we obtain

αi,j(λi − λj) = 0, 1 ≤ j ≤ n.

Since λi 6= λj , we get αi,j = 0 for i 6= j.Therefore, the derivations of the algebra L have the following form:

Der(L) =

0 β1 β2 . . . βn0 α1,1 0 . . . 00 0 α2,2 . . . 0. . . . . . . . . . . . . . .0 0 0 . . . αn,n


Let ∆ be a local derivation on L and let∆(x) = ξx+ ξ1e1 + ξ2e2 + · · ·+ ξnen,∆(ei) = ζix+ ζi,1e1 + ζi,2e2 + · · ·+ ζi,nen, 1 ≤ i ≤ n.

From the equalities ∆(x) = dx(x) and ∆(ei) = dei(ei) for 1 ≤ i ≤ n, itis easy to obtain, that

ξ = ζi = ζj,k = 0, 1 ≤ i, j, k ≤ n, j 6= k.

Consequently, we have ∆ ∈ Der(L), i.e., any local derivation on L is aderivation.

Case 2. Let λi = λj for some i and j. Without loss of generality, we canassume

λ1 = · · · = λs, λs+1 = · · · = λs+p, . . . λn−q = · · · = λn.

Similarly, to the case 1, using the property of derivation we describeall derivation of the algebra L and odtain that the general matrix form ofDer(L) has the form

0 β1 ... βs βs+1 ... βs+p ... βn−q ... βn0 α1,1 ... α1,s 0 ... 0 ... 0 ... 0... ... ... ... ... ... ... ... ... ... ...0 αs,1 ... αs,s 0 ... 0 ... 0 ... 00 0 ... 0 αs+1,s+1 ... αs+1,s+p ... 0 ... 0... ... ... ... ... ... ... ... ... ... ...0 0 ... 0 αs+p,s+1 ... αs+p,s+p ... 0 ... 0... ... ... ... ... ... ... ... ... ... ...0 0 ... 0 0 ... 0 ... αn−q,n−q ... αn−q,n... ... ... ... ... ... ... ... ... ... ...0 0 ... 0 0 ... 0 ... αn,n−q ... αn,n

It should be noted that all parameters of the previous matrix are free.

Thus, for any local derivation ∆ on L, considering ∆(x) and ∆(ei) for1 ≤ i ≤ n, analogously to the case 1, it is not difficult to show that ∆ is aderivation. Therefore, any local derivation on L is a derivation.

Now let adx be a not diagonalized operator on N . Then in this Jordanform

adx =

J1 0 . . . 00 J2 . . . 0. . . . . . . . . . . .0 0 ... Js


there exists a Jordan block with order k(k ≥ 2). Without loss of generalityone can suppose J1 has order k ≥ 2. Then the table of multiplication of thealgebra L on the basis x, e1, e2, . . . ek, ek+1, . . . en has the following form:

e1x = λ1e1 + e2,e2x = λ1e2 + e3,. . . . . . . . . . . .ekx = λ1ek,eix = λiei + µiei+1, k + 1 ≤ i ≤ n,

where µi = 0; 1.By the direct verification of the property of derivation we obtain that

the general form of the matrix of Der(L) has the form:

0 β1 β2 ... βk−1 βk βk+1 ... βn0 α1,1 α1,2 . . . α1,k−1 α1,k 0 ... 00 0 α1,1 ... α1,k−2 α1,k−1 0 ... 0... ... ... ... ... ... ... ...0 0 0 ... α1,1 α1,2 0 ... 00 0 0 ... 0 α1,1 0 ... 00 0 0 ... 0 0 H2 ... 0... ... ... ... ... ... ... ... ...0 0 0 ... 0 0 0 ... Hs

where Hi are block matrices with the same order of Jordan blocks Ji.

Consider the linear operator ∆ : L→ L by

∆(x) = 0, ∆(ei) = ei, 1 ≤ i ≤ k − 1,

∆(ek) = 2ek, ∆(ei) = 0, k + 1 ≤ i ≤ n.

It is easy to see that ∆ is not a derivation, but it is local derivation.Indeed, for any element y = γx+η1e1+η2e2+ · · ·+ηnen ∈ L we consider

∆(y) = ∆(γx+η1e1+η2e2+· · ·+ηnen) = η1e1+η2e2+· · ·+ηk−1ek−1+2ηkek.

Consider the derivation dy such that

dy(x) = 0, dy(ei) = 0, k + 1 ≤ i ≤ n,

dy(ei) = α1,1ei + α1,2ei+1 + · · ·+ α1,k−i+1ek, 1 ≤ i ≤ k.


Thendy(y) = dy(γx+ η1e1 + η2e2 + · · ·+ ηnen) =

= η1α1,1e1 +(η2α1,1 + η1α1,2)e2 + · · ·+(ηkα1,1 + ηk−1α1,2 + · · ·+ η1α1,k)ek.

Suppose that ∆(y) = dy(y). Then we get

η1 = η1α1,1,

η2 = η2α1,1 + η1α1,2,

. . . . . . . . .

ηk−1 = ηk−1α1,1 + ηk−2α1,2 + · · ·+ η1α1,k−1,

2ηk = ηkα1,1 + ηk−1α1,2 + · · ·+ η1α1,k.

Note that this system has a solution with respect to αi,j for anyparameters ηi. Indeed,

• if η1 6= 0, then

α1,1 = 1, α1,2 = · · · = α1,k−1 = 0, α1,k =ηkη1,

• if η1 = · · · = ηs−1 = 0 и ηs 6= 0, 2 ≤ s ≤ k − 1, then

α1,1 = 1, α1,2 = · · · = α1,k−s = 0, α1,k−s+1 =ηkηs,

• if η1 = · · · = ηk−1 = 0 и ηk 6= 0 then we have α1,1 = 2.

Thus, we proved that ∆ is a local derivation.

References

1. Ayupov Sh.А., Kudaybergenov К.К., Local derivations on finite-dimensional Lie algebras. Linear Alg. and Appl., 2016, Vol. 493, p. 381–388.

2. Albeverio S., Ayupov Sh.A., Kudaybergenov K.K., Nurjanov B.O., Localderivations on algebras of measurable operators. Comm. in Cont. Math.,2011, Vol. 13, No. 4, p. 643–657.

3. Bresar M., Semrl P., Mapping which preserve idempotents, localautomorphisms, and local derivations. Canad. J. Math. 1993, Vol. 45, p.483-496.

4. Jacobson N. Lie algebras, Interscience Publishers, Wiley, New York, 1962.


5. Kadison R.V, Local derivations. Journal of Algebra., 1990, Vol. 130, p. 494–509.

6. Mubarakzjanov G.M., On solvable Lie algebras (Russian), Izv. Vyss. Ucehn.Zaved. Matematika, 1963, Vol. 1, p. 114–123.

Khudoyberdiyev A.Kh.Institute of Mathematics, Uzbek Academy of Sciences MirzoUlugbek street, 81, 100170 Tashkent, Uzbekistan, e-mail:[email protected] D.Y.Department of Mathematics, National University of Uzbekistan,University street, 4, 100174, Tashkent, Uzbekistan, e-mail:sultonova-dilrabo@mail/ru

108 Khusanbaev Y.M., Sharipov S.O.

Uzbek MathematicalJournal, 2018, No 1 , pp.108-114

On convergence of branching processes with weaklydependent immigration

Khusanbaev Y.M., Sharipov S.O.

Abstract. In this paper we study the convergence of critical branchingprocesses with immigration to deterministic process assuming that theimmigration is φ− mixing.

Keywords: Branching processes, immigration, φ− mixing, Skorokhod space.Mathematics Subject Classification (2010): 60G10, 60J80

1 IntroductionLet Yki, k, i ∈ N and εk, k ∈ N be two sequences of independent non-negative integer-valued random variables, and Yki, k, i ∈ N are independent andidentically distributed. Define the sequence of random variables W (k) , k ≥ 0recursively as:

W (k) =

W (k−1)Xi=1

Yki + εk, k ≥ 1,W (0) = 0. (1.1)

The sequence W (k) , k ≥ 0 is called a branching process with immigration. Wecan interpret W (k) as the size of the k−th generation of a population and Yki

is the number of offsprings of the i−th individual in the (k − 1)st generationand εk is the number of immigrants contributing to the k−th generation. Assumethat a = EY11 < ∞. The sequence of branching process W (k) , k ≥ 0 is calledsubcritical, critical and supercritical if a < 1, a = 1 and a > 1, respectively. Suchprocesses have a number of applications in biology, finance, economics, queueingtheory etc.(see e.g.[10]).

Many papers devoted to study asymptotic behavior of process (1). Specialinterest represents an investigation of sufficient conditions of convergence ofproperly normalized process W ([nt]),t ≥ 0 as n → ∞ to non-random process,where [x] denotes integer part of x.For instance, Khusanbaev [1], [2] studied criticaland supercritical branching processes with immigration under assumption thatimmigration process εk, k ∈ N is independent. In papers [3],[4] this problem wasconsidered for branching processes with nearly critical immigration. Other resultsinclude functional limit theorem for properly normalized process W ([nt]),t ≥ 0(see [5], [6] and references therein).

The aim of this paper is to formulate sufficient condition of convergence ofprocess (1) to deterministic process in the case of φ− mixing immigration.

On convergence of branching processes with ... 109

2 PreliminariesRecall the notion of φ−mixing random variables.

Definition 2.1. A sequence of random variables Xn, n ≥ 1 is calledφ−mixing(or uniformly strong mixing) if

φ (n) = supk≥1,A∈Ξk1 ,P (A)>0,B∈Ξ∞

k+n

|P (B |A )− P (B)| → 0 as n→∞,

where Ξnm is the σ−field generated by the random variables Xm, Xm+1, ..., Xn.

The concept of φ−mixing dependence was introduced by Ibragimov [9].

The next lemma establish some bounds on the covariance cov (X,Y ) for theabove kind of mixing sequence.

Lemma 2.2. Let Xn, n ∈ Z be a φ− mixing sequence, X ∈ Lp

`Ξk

1

ánd Y ∈

Lq (Ξ∞k+n) with p, q ≥ 1 and 1p

+ 1q

= 1. Then

|EXY − EXEY | ≤ 2(φ (n))1/p(E|X|p)1/p(E|Y |q)1/q,

where Lp

`Ξb

a

ís the space of Ξb

a−measureable random variables with finite p−thmoment.

The Lemma 2.2 is due to Ibragimov [7]. Now for any n ≥ 0 , let = (n) be theσ−field generated by W (0) ,W (1) , ...,W (n). Denote α (n) = Eεn < ∞ andβ (n) = V ar (εn) < ∞. For each n ≥ 1, suppose α (n) and β (n) are regularlyvarying functions as n→∞, i.e.,

α (n) = nα`α (n) , β (n) = nβ`β (n) ,

where α, β ≥ 0, and `α (x), and `β (x) are slowly varying functions as n → ∞.Moreover, we assume that EY11 = 1, b2 = V arY11 <∞. If a sequence an, n ≥ 1is regularly varying with exponent δ, we will write an, n ≥ 1 ∈ <δ. So α (n) ∈<αand β (n) ∈ <β .

The following lemma is from Rahimov [5].

Lemma 2.3. If an, n ≥ 1 ∈ <δ, then for any τ ∈ (−δ − 1,∞)

nXk=1

kτak ∼nτ+1an

τ + δ + 1, as n→∞,

and

nPk=1

kτak, n ≥ 1

ff∈ <δ+τ+1 .

In the sequel a ∼ b denotes lim ab

= 1, a = o (b) denotes lim ab

= 0, andP→

denotes convergence in probability. C denotes a generic constant which mighthave different values in different inequalities, but does not depend on n.


3 Main result and its proofOur main result is the following theorem.

Theorem 3.1. Assume that the following conditions are fulfilled:the sequence of immigration process εn, n ≥ 1 is φ−mixing with

∞Xn=1

φ1/2 (n) <∞; (3.1)

α (n) →∞, β (n) = o (nα (n)) as n→∞. (3.2)

Then for any T > 0 the random process W ([nt])

n1+α`α(n), t ∈ [0, T ] converges weakly to

µ (t), t ∈ [0, T ] in Skorokhod space D [0, T ], where the process µ (t) is defined asµ (t) = t1+α

1+α, t ≥ 0.

Proof. Clearly, in order to prove the theorem it is enough to show that forany T > 0

sup0≤t≤T

˛W ([nt])

n1+α`α (n)− t1+α

1 + α

˛P→ 0 as n→∞. (3.3)

We will prove (3.3). One can easily verify that W (k) can be rewritten as

W (k) = W (k − 1) + α (k) +M1 (k) +M2 (k) , (3.4)

where M1 (k) =W (k−1)P

j=1

(Ykj − 1), M2 (k) = εk − α (k).

Adding identities (3.4), we obtain

W ([nt]) =

[nt]Xk=1

α (k) +

[nt]Xk=1

M1 (k) +

[nt]Xk=1

M2 (k). (3.5)

Since Yki, k, i ∈ N are independent random variables and M1 (k),k ≥ 1 areindependent random variables with respect to the fixed σ−field = ([nt]) then byconditional inequality of Kolmogorov

P

0@ 1

n1+α`α (n)sup

0≤t≤T

˛˛ [nt]Xk=1

M1 (k)

˛˛ > ε

1A ≤

≤ 1

ε2n2(1+α)`2α (n)

24b2 [nT ]Xk=1

EW (k − 1)

35 . (3.6)


Again by (3.4) and recurrence we get EW (k) =kP

i=1

α (i). Therefore

[nT ]Xk=1

EW (k) =

[nT ]Xk=2

k−1Xi=1

α (i) = [nT ]

[nT ]Xk=1

α (k)−[nT ]Xk=1

kα (k). (3.7)

Now from Lemma 2.3, (3.2) and using the properties of regularly varying functions,one can get

[nT ]Xk=1

α (k) ∼ nα (n)

1 + αT 1+α,

[nT ]Xk=1

kα (k) ∼ n2α (n)

2 + αT 2+α, (3.8)

[nT ]Xk=1

β (k) ∼ nβ (n)

β + 1T 1+β as n→∞.

Consequently, by the Chebyshev inequality and together with (3.6) and (3.7), wehave

P

0@ 1

n1+α`α (n)sup

0≤t≤T

˛˛ [nt]Xk=1

M1 (k)

˛˛ > ε

1A ≤

≤ b2T 1+α

ε2n2α`α (n)

„nα

1 + α+

1

2 + αT

«→ 0 as n→∞. (3.9)

Now, clearly, εk−α (k),k ∈ N satisfies φ−mixing condition with the same mixingcoefficients as εk,k ∈ N. From the Lemma 2.2 we have for i < j

|cov (εi, εj)| ≤ 2φ1/2 (j − i)β1/2 (i)β1/2 (j) .

Then by (3.7) and using properties of regularly varying functions we get

E

24 [nt]Xk=1

M2 (k)

352

=

[nt]Xk=1

β (k) + 2

[nt]Xk=2

k−1Xj=1

cov (εk, εj) ≤

≤

1 + 4

∞Xk=1

φ1/2 (k)

![nt]Xk=1

β (k) ∼ Cnβ (n) t1+β , n→∞. (3.10)

By the Chebyshev’s inequality and taking into account assumptions of theorem,one has

P

0@ 1

n1+α`α (n)

˛˛ [nt]Xk=1

M2 (k)

˛˛ > ε

1A ≤ 1

ε2n2(1+α)`2α (n)E

24 [nt]Xk=1

EM2 (k)

352

∼


∼ Cnβ (n) t1+β

ε2(nα (n))2→ 0 as n→∞.

Hence, for any t ∈ [0, T ]

1

n1+α`α (n)

[nt]Xk=1

M2 (k)P

→ 0 as n→∞. (3.11)

With the same arguments we obtain

1

n2(1+α)`2α (n)E

˛˛ [nt]Xk=[ns]+1

M2 (k)

˛˛2

≤ C“t1+β − s1+β

”,

where 0 < s < t < ∞. Then by (3.11) and Theorem 15.6 from [8] we conclude

that the process 1n1+α`α(n)

[nt]Pk=1

M2 (k), t ≥ 0 converges weakly to zero in Skorokhod

space D [0, T ]. Hence

1

n1+α`α (n)sup

0≤t≤T

˛˛ [nt]Xk=1

M2 (k)

˛˛ P

→ 0 (3.12)

as n→∞. By (3.8) we deduce that

1

n1+α`α (n)

[nt]Xk=1

α (k) ∼ 1

n1+α`α (n)

nα (n)

1 + αt1+α → t1+α

1 + α, n→∞. (3.13)

Finally, collecting (3.5), (3.9), (3.12) and (3.13), we get (3.3) which proves theTheorem 3.1.

Remark 3.2. If εk, k ∈ N is a sequence of independent random variables thenconclusion of theorem coincides with the results of [1].

Remark 3.3. In [6], the functional limit theorem was obtained for fluctuationof process W ([nt]),t ≥ 0 under assumption that εk, k ∈ N is a sequenceof m− dependent random variables. It is well-known that for m− dependentrandom variables condition (3.1) of the Theorem 3.1 holds. Consequently, ourresult improves the known results.

From Theorem 3.1 and continuous mapping theorem we can formulate thefollowing corollary on the behaviour of total number of a population.

Corollary 3.4. Under assumptions of Theorem 3.1 we have for any t ≥ 0

1

n2+α`α (n)

[nt]Xk=1

W (k)P→ t2+α

(1 + α) (2 + α)as n→∞.


As an example, we give the following corollary.

Corollary 3.5. Assume εk, k ≥ 1 are Poisson random variables satisfying

φ−mixing condition with∞P

k=1

φ1/2 (k) < ∞ and with mean α (k). If α (k) =

kα`α (k), α ≥ 0 and α (k) →∞ as k →∞ then clearly β (k) = o (kα (k)),k →∞.Thus, all conditions of Theorem 3.1 are satisfied and corollary can be obtainedfrom Theorem 3.1.

References

1. Y.M.Khusanbaev, On asymptotic of critical branching processes with non-homogeneous and increasing immigration, (in Russian). Doklady Academyof Sciences of Uzbekistan. 3 (2010), 6–10.

2. Y.M.Khusanbaev, On asymptotic of subcritical branching process withimmigration,(in Russian). Ukrainian Mathematical Journal. 65(6) (2013),835–843.

3. Ispany M., Pap G., Van Zuijlen M.C.A, Fluctuation limit theoremof branching processes with immigration and estimation of the mean,Adv.Appl.Probab, 37 (2005), 523–538.

4. Y.M.Khusanbaev, Nearly critically branching processes and limittheorems,(in Russian). Ukrainian Mathematical Journal. 61(1) (2009),127–133.

5. I.Rahimov, Functional limit theorems for a critical branching processes withimmigration, Adv.Appl.Probab. 39 (2007), 1054–1069.

6. Guo H., Zhang M., A fluctuation limit theorem for a critical branchingprocess with dependent immigration,Statistics and Probability Letters. 94(2014), 29–38.

7. I.A.Ibragimov., Yu.V.Linnik, Independent and stationary sequences ofrandom variables, Wolters-Noordhoff. Groningen, (1971).

8. P.Billingsley, Convergence of probability measures. New York, Wiley. (1968).

9. I.A.Ibragimov, Some limit theorems for stationary processes, TheoryProbab.Apply. 7 (1962), 349–382.

10. Haccou P.,Jagers P., Vatutin V, Branching processes.Variation, Growth, andExtinction of Populations, Cambridge University Press. Cambridge. (2005).


Khusanbaev Y.M.Department of Theory of Probability and MathematicalStatistics, Institute of Mathematics, Uzbek Academy ofSciences, Mirzo Ulugbek street, 81, 100170, Tashkent,Uzbekistan, e-mail: [email protected] S.O.Department of Theory of Probability and MathematicalStatistics Institute of Mathematics,Uzbek Academy of Sciences,Mirzo Ulugbek street, 81, 100170, Tashkent, Uzbekistan, e-mail:[email protected]

Leibniz algebras corresponding to the Virasoro algebra 115


Leibniz algebras corresponding to the Virasoro algebraKurbanbaev T.K.

Abstract. In this work we investigate Leibniz algebras, whose quotient Liealgebra is the Virasoro algebra. We introduce a module for the Virasoro algebraand consider the Leibniz algebra L, whose the corresponding Lie algebra L/I isthe Virasoro algebra with the condition that the ideal I is module of the Virasoroalgebra.

Keywords: Leibniz algebra, Virasoro Lie algebra, Leibniz representation,right Lie module, classification

Mathematics Subject Classification (2010): 17A32, 17B30, 17B10

1 IntroductionLeibniz algebra is a non-associative algebra with a bilinear product satisfying theso-called the Leibniz identity, which converts right multiplication by an elementof the algebra in the derivation of this algebra. For the first time this classof the algebras was defined in the work of Bloh [2], but the active interest inthem appeared after the works of J.-L.Loday and T. Pirashvili, and namely,J.-L.Loday first used the term "Leibniz algebra"[6]. It is known that any Liealgebra is, in particular, the Leibniz algebra, and, therefore, many properties forLie algebras, continue to the case of Leibniz algebras. From the classical theoryof Lie algebras it is known that an arbitrary finite-dimensional Lie algebra over afield of characteristic zero decomposes into a semi-direct sum of a maximal solvableideal and its semisimple subalgebra [4]. The problem of classifying simple finite-dimensional Lie algebras over the field of complex numbers was solved by the endof the 19th century by V.Killing and E.Cartan [3], [4]. And ten years later E.Cartanclassified simple infinite-dimensional Lie algebras of vector fields on a finite-dimensional space. Starting with the works of Lie, V.Killing and E.Cartan, thetheory of finite-dimensional Lie groups and Lie algebras systematically developedin depth and breadth. On the other hand, E.Cartan’s works on simple infinite-dimensional Lie algebras were actually forgotten until the middle of the sixtiesof the 20th century. Interest in this field of mathematics began to revive in 1964with the works of V.W.Guillemin, S.Sternberg and I.M.Singer, who developed thecorresponding algebraic language and the technique of filtered and graded Liealgebras. Note, that in the works of V.Kac, infinite-dimensional Lie algebras andtheir representation [5] are studied. It is known that every non-Lie Leibniz algebraL contains a nontrivial ideal (denoted by I), which is subspace spanned by the

116 Kurbanbaev T.K.

squares of the elements of the algebra

I = span[x, x], x ∈ L.

This ideal I is a minimal ideal with the property that the quotient algebra L/Iis a Lie algebra. One approach to the study of Leibniz algebras is the descriptionof such algebras, whose quotient algebra with respect to the ideal I is a given Liealgebra. In particular, in work of B.A.Omirov a description of finite-dimensionalcomplex Leibniz algebras, whose quotient algebra is isomorphic to the simple Liealgebra sl2, are received [7].

D.Barnes proved an analogue of Levi’s theorem, namely, the finite-dimensionalLeibniz algebras also decompose into the semi-direct sum of a maximal solvableideal and a semi-simple Lie algebra [1]. Hence, we conclude that if the quotientalgebra of a Leibniz algebra is isomorphic to a semi-simple Lie algebra, thengiven a module over this semi-simple Lie algebra we can rebuilt the initial Leibnizalgebra.

In this paper we investigate Leibniz algebras, whose quotient Lie algebra is theVirasoro algebra. We introduce a module for the Virasoro algebra and considerthe Leibniz algebra L, whose the corresponding Lie algebra L/I is the Virasoroalgebra with the condition that the ideal I is module of the Virasoro algebra.

2 PreliminariesDefinition 2.1. An algebra (L, [·, ·]) over a field F is called a Leibniz algebra ifit is defined by the identity

[x, [y, z]] = [[x, y], z]− [[x, z], y], for all x, y ∈ L,

which is called Leibniz identity.

Let A = C[z, z−1] be the algebra of Laurent polynomials in one variable andconsider

Der(A) = f(z)d

dz: f ∈ C[z, z−1] = spanej := −zj+1 d

dz: j ∈ Z.

This is called the Witt algebra over C, denoted W ; its basis is ej : j ∈ Z and itsstructure is

[ei, ej ] = (i− j)ei+j .

TheVirasoro algebra (denoted by V ir) is called such algebra with a basisei, c : i ∈ Z and multiplication

[ei, ej ] = −[ej , ei] = (i− j)ei+j + δi,−ji3 − i

12c, [ei, c] = 0,


where δi,−j is the Kronecker symbol, that is

δi,−j =

(1, if i = −j,0, if i 6= −j.

The Virasoro algebra is the one-dimensional central extension of the Witt algebra.Let V be countably dimensional vector space and

V = V (α, β) = P (z)zα(dz)β ∼= C[z, z−1] = spanv(n) : n ∈ Z,

where α, β ∈ C and P ∈ C[z, z−1]. We define a module of the Virasoro algebra bythe following multiplication:

[v(k), en] = (k + α+ βn)v(n+ k), (2.1)

where en := −zn+1 ddz, v(k) := zk+α(dz)β , α, β ∈ C, n, k ∈ Z.

Theorem 2.2. [5] The module V (α, β) of V ir is reducible if

(i) α ∈ Z and β = 0, or (ii) α ∈ Z and β = 1;

otherwise V (α, β) is irreducible.

3 Main resultsLet G be the Virasoro algebra and the vector space V = V (α, β) = v(n) | n ∈ Zbe G-module defined by (2.1). Then we will construct the Leibniz algebra L =G⊕V and assume that I = V is an ideal and L/I = G. So we have multiplication(2.1) and [G,V ] = [V, V ] = 0, [G,G] ⊆ G+V. In order to describe Leibniz algebrasL = G+ V , we need to consider the product [G,G].

In the following theorem we study the Leibniz algebra for α /∈ Z.

Theorem 3.1. Let L be a Leibniz algebra, whose quotient algebra is the Virasoroalgebra G. If α /∈ Z, then there exists a basis ei, c, v(i) | i ∈ Z in L such that Lis isomorphic to the following algebra:8><>:

[v(k), ei] = (k + α+ bi)v(i+ k), i, k ∈ Z,

[ei, e−i] = 2ie0 + i3−i12

c, i 6= 0,

[ei, ej ] = (i− j)ei+j , i 6= j,−j, 0,

and [G,G] ⊆ G.

118 Kurbanbaev T.K.

Proof.Let

[ei, ej ] = (i− j)ei+j +X

k

γi,j,kv(k), i 6= −j,

[ej , e−j ] = 2je0 +j3 − j

12c+

Xk

γj,−j,kv(k), j 6= 0,

[ej , ej ] =X

k

γj,j,kv(k), [ej , c] =X

k

λj,kv(k),

[c, ej ] =X

k

µj,kv(k), [c, c] =X

k

ηkv(k),

where α+ j + βj /∈ Z,∀j ∈ Z, α, β ∈ C.We have the following change of the basis

e′0 = e0 −X

k

γ0,0,k

α+ kv(k), e′s = es +

Xk

γs,0,k

s− α− kv(k),

c′ = c−X

k

µ0,k

α+ kv(k)

and get[c, e0] = 0, [ej , e0] = jej , j ∈ Z.

We consider the Leibniz identity for the following triples of the elements ofthe algebra and obtain the following restrictions:

e0, ej , e0 γ0,j,k = 0 and [e0, ej ] = −jej , j ∈ Z,c, c, e0 ηk = 0 and [c, c] = 0,

ej , ej , e0 γj,j,k = 0 and [ej , ej ] = 0,

ej , c, e0 λj,k = 0 and [ej , c] = 0,

c, ej , e0 µj,k = 0 and [c, ej ] = 0,

ej , e0, e−j γj,−j,k = 0, j, k ∈ Z and [ej , e−j ] = 2je0 + j3−j12

c, j 6= 0,

ei, e0, ej γi,j,k = 0, and [ei, ej ] = (i− j)ei+j

Therefore, we get [G,G] ⊆ G.Now let us consider the case α ∈ Z.

Theorem 3.2. Let L be a Leibniz algebra, whose quotient algebra is the Virasoroalgebra G. If α ∈ Z, then there exists a basis ei, c, v(i) | i ∈ Z in L such that Lis isomorphic to one of the following algebras:

(1) In the case α ∈ Z and β 6= −1, 1, 3 we obtain [G,G] ⊂ G and

(I) :

8><>:[v(k), ei] = (k + α+ bi)v(i+ k),

[ei, e−i] = 2ie0 + i3−i12

c, i 6= 0,

[ei, ej ] = (i− j)ei+j , i 6= j,−j, 0,


(2) In the case α ∈ Z and β−1, 1, 3 we have [G,G] ⊂ G or [G,G] ⊂ G+ Vand(a) β = −1 :

(II) :

8>>>><>>>>:[v(k), ei] = (α+ k − i)v(k + i),

[ei, ei] = bi,iv2i− α, i 6= 0,

[ei, e−i] = 2ie0 + i3−i12

c− 13bi,iv(−α), i 6= 0,

[ei, ej ] = (i− j)ei+j + bi,jv(i+ j − α), i 6= j,−j,

where

8>>>>>><>>>>>>:

b2,2 = 2, b−2,−2 = −2;

bi,i = 1(2i+1)(i−2)

((i+ 1)(2i− 3)bi−1,i−1 − (2i− 1)), i 6= −2,−1, 0, 1, 2;bi,i+1 = i+1

(2i+1)(i−1)((2i− 1)bi,i − 1);

bi,j = ji(j−2)

((i− 1)bi+1,j−1 + (j − i− 1)), i 6= j,−j,i, j 6= −1, 0, 1, j 6= i+ 1.

(b) β = 1 :

(III) :

8>>>>>>>>><>>>>>>>>>:

[v(k), ei] = (k + α+ i)v(i+ k),

[e0, ei] = −iei, i 6= 0,

[ei, e0] = iei, i 6= 0,

[ei, ei] = (i3 − i)v(2i− α), i 6= 0,

[ei, e−i] = 2ie0 + i3−i12

c+ (i3 − i)v(−α), i 6= 0,

[ei, ej ] = (i− j)ei+j + j(ij − 1)v(i+ j − α), i 6= j,−j, 0, j 6= 0,

(c) β = 3 :

(IV ) :

8>>>><>>>>:[v(k), ei] = (k + α+ 3i)v(i+ k),

[ei, ei] = bi,iv(2i− α), i 6= 0,−1, 1,

[ei, e−i] = 2ie0 + i3−i12

c− 13bi,iv(−α), i 6= 0,−1, 1,

[ei, ej ] = (i− j)ei+j + bi,jv(i+ j − 1), i 6= j,−j,

120 Kurbanbaev T.K.

where8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:

b2,2 = 9, b−2,−2 = −9;

bi,i =(i+ 1)(2i+ 1)bi−1,i−1 − (i+ 1)(4i− 3)ai−1

(i− 2)(2i− 3)−

− (4i− 1)(i− 2)a−i

(i− 2)(2i− 3), i 6= −2,−1, 0, 1, 2;

bi,i+1 =(i+ 1)((2i+ 3)bi,i + (4i+ 1)ai)

(i− 1)(2i+ 1), i 6= −1, 0, 1;

bi,j =j((i− 1)bi+1,j−1 + (3i+ j)(j − 1)aj−1)

i(j − 2)−

− j((j − i− 1)(j + i− 1)aj+i−1 − i(i+ 3j − 2)ai)

i(j − 2),

j 6= i+ 1, i 6= j,−j, i, j 6= −1, 0, 1;

ai =(i− 1)(i+ 2)(i+ 3)

20, a−i = − (i+ 1)(i− 2)(i− 3)

20, i 6= −1, 0, 1.

The theorem is proved analogously to the 3.1.

References

1. Barnes D., On Levi Theorem for Leibniz algebra, Bull. of the AustralianMath. Soc., 86 (2012), 184-185.

2. Bloh A., A generalization of the conspekt of a Lie algebra, Sov.Math.Dokl.,6 (1965) 1450-1452.

3. Humphreys J.E., Introduction to Lie algebras and Representation theory,Springer-Verlag New York, 1972. p 186.

4. Jecobson N., Lie algebras, 340p. Interscience Publishers, Wiley, New York,(1962).

5. Kac V., Raina A., Bombay lectures on highest weight representations ofinfinite dimensional Lie algebras, World Sci.Singapore, 1987.

6. Loday J.-L. Une version non commutative des algebres de Lie: les algebresde Leibniz, Enseign.Math., (2) 39 (3-4) (1993) 269-293.

7. Omirov B.A., Rakhimov I.S., Turdibaev R.M., On description of Leibnizalgebras corresponding to sl2, Algebr.Represent.Theory. 16(5), 2013, 1507-1519.

Kurbanbaev T.K.Institute of Mathematics, Uzbek Academy of Sciences MirzoUlugbek street, 81, 100170 Tashkent, Uzbekistan,e-mail: [email protected]

Estimation of the mean of spatial autoregressive ... 121

Uzbek MathematicalJournal, 2018, No 1 , pp.121-127

Estimation of the mean of spatial autoregressiveprocesses in Banach space

Kushmurodov A.A., Sharipov O.Sh.

Abstract. As an estimator of the unknown expectation of the spatialautoregressive process with values in Banach space we take a sample mean. Astrong consistency of this estimator is proven in this paper.

Keywords: Banach space,random fields,spatial autoregressive processes,sample mean

Mathematics Subject Classification (2010): 60F15,60G60,60G50

1 IntroductionWe consider an autoregressive process in a separable Banach space B (with anorm‖·‖ ) defined as

Xn,j − µ = ρ1 (Xn,j−1 − µ) + ρ2 (Xn−1,j − µ) + ρ3 (Xn−1,j−1 − µ) + εn,j , (1.1)

(n, j) ∈ Z2, ρi : B → B are bounded and linear operators with the operatornorm ‖·‖L and µ ∈ B.

˘εn,j , (n, j) ∈ Z2

¯is an innovation process.

Real-valued spatial autoregressive processes of type (1.1) were studied in [1].In[5],[6] linear random fields were investigated.

Spatial autoregressive processes (1.1) were studied in [3],[4].In [4] it is proventhat (1.1) has a unique stationary solution if

˘εn,j , (n, j) ∈ Z2

¯is stationary and

ρi satisfy some conditions.We assume that B is Rademacher type p (1 ≤ p ≤ 2).Definition. We say that a separable Banach space B (with a norm ‖·‖ ) is of

Rademacher type p (1 ≤ p ≤ 2) if there exists a constant 0 < C < ∞ dependingonly on B, such that

E

‚‚‚‚‚nX

i=1

Xi

‚‚‚‚‚p

≤ C

nXj=1

E ‖Xj‖p,

for every finite collection X1, X2, ..., Xn of independent mean zero randomvariables with values in B and E ‖Xj‖p <∞ j = 1, 2, ....

Note that Lp,lp (1 ≤ p ≤ 2) spaces are Rademacher type p spaces and a Hilbertspace is Rademacher type 2 space.

As an estimator of the mean µ we take a sample mean Xm,n = 1mn

mPi=1

nPj=1

Xi,j .

Without loss of generality we suppose that µ = 0. In the next theorem we establisha strong consistency of the Xm,n.

122 Kushmurodov A.A., Sharipov O.Sh.

Theorem 1.1. Let˘Xn,j , (n, j) ∈ Z2

¯be a spatial autoregressive process with

values in Rademacher type p (1 < p ≤ 2) Banach space B satisfying (1.1). Assumethat the innovation process and operators satisfy the following conditions:

‖ρi‖L < 1, i = 1, 2, 3, ‖ρ1 + ρ2 + ρ3‖L < 1, (1.2)

Eε(i, j) = 0, E‖ε(i, j)‖p < ∞, (i, j) ∈ Z2. (1.3)

ThenXm,n → 0, a.s. m, n→∞.

In the next theorem by H we denote a separable Hilbert space with innerproduct (·,·) and a norm ‖·‖ .

Theorem 1.2. Let˘Xn,j , (n, j) ∈ Z2

¯be an autoregressive process in H

satisfying (1.1)-(1.2). Assume that the innovation process satisfies the followingconditions:

Eε(i, j) = 0, (i, j) ∈ Z2, sup(i,j)∈Z2

E‖ε(i, j)‖2 < M, (1.4)

sup(i,j)∈Z2

|E (ε(i, j), ε(i+m, j + l))| ≤ ϕ`‖(m, l)‖1

´,

mXi=1

nXj=1

ϕ`‖(i, j)‖1

´= o(nm), m, n→∞,

for some non-increasing function φ(·) , M > 0 and a norm ‖·‖1 in Z2.Then

Xm,n → 0, a.s., as m, n→∞.

Autoregressive processes (non-spatial) with values in Banach spaces wereinvestigated in [7],[8].

2 Preliminary resultsWe will use the following theorems in the proofs of Theorems 1.1 and 1.2 Thesetheorems were proved in [2].

Theorem 2.1. Let X(i, j), (i, j) ∈ Z2 be a random field of independent randomvariables with values in a separable Banach space B of Rademacher type p (1 <p ≤ 2) with

EX(i, j) = 0 (i, j) ∈ Z2, supi,j

E‖X(i, j)‖p < C.


Then as m ∨ n→∞,

(mn)γ1

(logmn)β

nPi=1

mPj=1

X(i, j)

mn→ 0, a.s.,

where γ1 = 1− 1p

and β > 1p, m ∨ n = max(m,n).

In Hilbert space case we will assume that X(i, j) satisfies the followingconditions:

EX(i, j) = 0, supi,j

E‖X(i, j)‖2 < M, (2.1)

supi,j

|E (X(i, j), X(i+m, j + l))| ≤ ϕ`‖(m, l)‖1

´, (2.2)

for some non-increasing function φ(·) , M > 0 and a norm ‖·‖1 in Z2.

Theorem 2.2. Let X(i, j) be a random field with values in H satisfying theconditions (2.1),(2.2) and:

nXi=1

mXj=1

ϕ`‖(i, j)‖1

´= o(nm), as, m ∨ n→∞.

Then as m ∨ n→∞, for some γ ∈ [1, 2) and β > 12,

(mn)2−γ

4

(logmn)β

1

mn

nXi=1

mXj=1

X(i, j) → 0, a.s..

where m ∨ n = max(m,n).

Proofs of Theorems.

Proof of Theorem 1.2In order to prove the theorem we need a representation for Xm,n.From (1.1) we have

mXk=1

nXl=1

X (k, l) =

m−1Xk=0

nXl=1

ρ1 (X (k, l)) +

mXk=1

n−1Xl=0

ρ2 (X (k, l)) +

+

m−1Xk=0

n−1Xl=0

ρ3 (X (k, l)) +

mXk=1

nXl=1

ε (k, l) =

=

mXk=1

nXl=1

ρ1 (X (k, l)) +

nXl=1

ρ1 (X (0, l))−nX

l=1

ρ1 (X (m, l)) +


+

mXk=1

nXl=1

ρ2 (X (k, l)) +

mXk=1

ρ2 (X (k, 0))−mX

k=1

ρ2 (X (k, n)) +

+

mXk=1

nXl=1

ρ3 (X (k, l)) +

mXk=1

ρ3 (X (k, 0))−mX

k=1

ρ3 (X (k, n)) +

+

nXl=1

ρ3 (X (0, l))−nX

l=1

ρ3 (X (m, l)) + ρ3 (X (0, 0))− ρ3 (X (m, 0))−

−ρ3 (X (0, n)) + ρ3 (X (m,n)) +

mXk=1

nXl=1

ε (k, l)

The latter implies

mXk=1

nXl=1

X (k, l)−mX

k=1

nXl=1

ρ1 (X (k, l))−mX

k=1

nXl=1

ρ2 (X (k, l))−mX

k=1

nXl=1

ρ3 (X (k, l)) =

= (ρ1 + ρ3) (

nXl=1

X (0, l))+ (ρ2 + ρ3) (

mXk=1

X (k, 0))− (ρ1 + ρ3) (

nXl=1

X (m, l))−

− (ρ2 + ρ3) (

mXk=1

X (k, n)) + ρ3 (X (0, 0))− ρ3 (X (m, 0))− ρ3 (X (0, n)) +

+ρ3 (X (m,n)) +

mXk=1

nXl=1

ε (k, l)

Now we have

(I − ρ1 − ρ2 − ρ3) (

mXk=1

nXl=1

X (k, l)) = (ρ1 + ρ3)(

nXl=1

X (0, l))+

+ (ρ2 + ρ3) (

mXk=1

X (k, 0))− (ρ1 + ρ3) (

nXl=1

X (m, l))− (ρ2 + ρ3) (

mXk=1

X (k, n))+

+ρ3 (X (0, 0))− ρ3 (X (m, 0))− ρ3 (X (0, n)) + ρ3 (X (m,n)) +

mXk=1

nXl=1

ε (k, l)

Finally we have the following representation

mXk=1

nXl=1

X (k, l) = (I − ρ1 − ρ2 − ρ3)−1 (ρ1 + ρ3) (

nXl=1

X (0, l))+

+(I − ρ1 − ρ2 − ρ3)−1 (ρ2 + ρ3) (

mXk=1

X (k, 0))−


−(I − ρ1 − ρ2 − ρ3)−1 (ρ1 + ρ3) (

nXl=1

X (m, l))−

−(I − ρ1 − ρ2 − ρ3)−1 (ρ2 + ρ3) (

mXk=1

X (k, n))+

+(I − ρ1 − ρ2 − ρ3)−1ρ3 (X (0, 0))− (I − ρ1 − ρ2 − ρ3)−1ρ3 (X (m, 0))−

−(I − ρ1 − ρ2 − ρ3)−1ρ3 (X (0, n)) + (I − ρ1 − ρ2 − ρ3)−1ρ3 (X (m,n)) +

+(I − ρ1 − ρ2 − ρ3)−1(

mXk=1

nXl=1

ε (k, l))

Thus the sample mean can be represented as

Xm,n =(I − ρ1 − ρ2 − ρ3)−1

mn(ρ1 + ρ3) (

nXl=1

X (0, l))+

+(I − ρ1 − ρ2 − ρ3)−1

mn(ρ2 + ρ3) (

mXk=1

X (k, 0))−

− (I − ρ1 − ρ2 − ρ3)−1

mn(ρ1 + ρ3) (

mXl=1

X (n, l))−

− (I − ρ1 − ρ2 − ρ3)−1

mn(ρ2 + ρ3) (

mXk=1

X (k, n))−

− (I − ρ1 − ρ2 − ρ3)−1

mnρ3 (X (0, n))− (I − ρ1 − ρ2 − ρ3)−1

mnρ3 (X (m, 0)) +

+(I − ρ1 − ρ2 − ρ3)−1

mnρ3 (X (0, 0)) +

(I − ρ1 − ρ2 − ρ3)−1

mnρ3 (X (m,n)) +

+(I − ρ1 − ρ2 − ρ3)−1

mn(

mXk=1

nXl=1

ε (k, l)) (2.3)

Now we will estimate each summand using Minkowski’s inequality

P

0BB@‚‚‚‚‚‚‚‚

nPl=1

X (0, l)

mn

‚‚‚‚‚‚‚‚ ≤ ε

1CCA ≤E

‚‚‚‚ nPl=1

X (0, l)

‚‚‚‚p

(mn)pεp≤

„E

‚‚‚‚ nPl=1

X (0, l)

‚‚‚‚p« 1p

!p

(mn)p ≤

≤ C1np

(mn)p =C1

mp= O

„1

mp

«(2.4)


By the same way we have

P

0BB@‚‚‚‚‚‚‚‚

mPk=1

X (k, 0)

mn

‚‚‚‚‚‚‚‚ ≥ ε

1CCA ≤E

‚‚‚‚ mPk=1

X (k, 0)

‚‚‚‚p

(mn)pεp≤

„E

‚‚‚‚ mPk=1

X (k, 0)

‚‚‚‚p« 1p

!p

(mn)p ≤

≤ C1mp

(mn)p =C1

np= O

„1

np

«(2.5)

P

0BB@‚‚‚‚‚‚‚‚

nPl=1

X (m, l)

mn

‚‚‚‚‚‚‚‚ ≥ ε

1CCA ≤E

‚‚‚‚ nPl=1

X (m, l)

‚‚‚‚p

(mn)pεp≤

„E

‚‚‚‚ nPl=1

X (m, l)

‚‚‚‚p« 1p

!p

(mn)p ≤

≤ C1np

(mn)p =C1

mp= O

„1

mp

«(2.6)

P

0BB@‚‚‚‚‚‚‚‚

mPk=1

X (k, n)

mn

‚‚‚‚‚‚‚‚ ≥ ε

1CCA ≤E

‚‚‚‚ mPk=1

X (k, n)

‚‚‚‚p

(mn)pεp≤

„E

‚‚‚‚ mPk=1

X (k, n)

‚‚‚‚p« 1p

!p

(mn)p ≤

≤ C1mp

(mn)p =C1

np= O

„1

np

«(2.7)

Using Markov’s inequality we get

P

„‚‚‚‚ρ3 (X (o, n))

mn

‚‚‚‚ > ε

«≤ E‖ρ3 (X (0, n))‖p

(mn)pεp≤ C

(mn)p = O

„1

(mn)p

«(2.8)

It remains to apply the Theorem 2.1 to the last summand and get

1

mn

mXk=1

nXl=1

ε (k, l) → 0 a. s., (2.9)

which implies

(I − ρ1 − ρ2 − ρ3)−1

1

mn

mXk=1

nXl=1

ε (k, l)

!→ 0 a. s.

Relations (2.4)-(2.9) and Borel-Cantelli lemma complete the proof of Theorem1.1.

The proof of Theorem 1.2 is almost the same as the proof of Theorem 1.1.We use the representation (2.3)and estimate each summand as in the proof ofTheorem 1.1 using the condition (1.4) instead of (1.3). The convergence of thelast summand to zero a.s. follows from Theorem 2.2. The theorem is proven.


References

1. Martin R.J., A subclass of lattice processes applied to a problem in planarsampling. Biometrika, 1979. V.66. pp. 209-217.

2. Sharipov O.,Sh., Kushmurodov A.A., Strong laws of large numbersfor random fields with values in Banach spaces.Uzbek mathematicaljournal,2017. No.1,165-171.

3. Alvares-Liebana J.,Bosq D.,Ruiz-Medina M.D., Asymptotic propertiesof a component-wise ARH(1) plug-in predictor.Journal of MultivariateAnalysis,2017. 155. pp .12-34.

4. Ruiz-Medina M.D., Spatial autoregressive and moving average Hilbertianprocesses.Journal of Multivariate Analysis,2011. 102.pp. 292-305.

5. Paulauskas V., On Beveridge-Nelson decomposition and limit theorems forlinear rabdom fields.Journal of Multivariate Analysis,2010. 101. pp.621-639.

6. Povilas B.,Youri D., Paulauskas V., Remarks on the SLLN for linear randomfields.Stat. and Probab. Letters, 2010,v.80, issue5-6,489-496.

7. Bosq D., Linear Processes in Function Spaces.Theory and applications.Lecture notes in Statistics, 149, Springer, 2000.

8. Dehling H., Sharipov O.Sh., Estimation of mean and covarianceoperator for Banach space valued autoregressive processes with dependentinnovations.Statistical Inference for Stochastic processes, 2005.v.8, 137-149.

Kushmurodov A.A.Institute of Mathematics,Uzbek Academy of science MirzoUlugbek street, 81, 100170, Tashkent, Uzbekistan, e-mail:[email protected] O.Sh.Institute of Mathematics,Uzbek Academy of science MirzoUlugbek street, 81, 100170, Tashkent, Uzbekistan, e-mail:[email protected]

128 Nuraliev F.A.


Optimal interpolation formulas in the space L(4)2 (0, 1)

Nuraliev F.A.

Abstract. In the present paper using S.L.Sobolev’s method optimalinterpolation formulas are constructed in L

(4)2 (0, 1) space. The explicit formulas

for coefficients of optimal interpolation formulas are obtained.Keywords: ptimal interpolation formula, optimal coefficients, S.L. Sobolev’s

method, the error functionalMathematics Subject Classification (2010):

1 IntroductionThe first spline functions were bonded from pieces of cubic polynomials. Further,this construction was modified, degree of polynomial was increased, boundaryvalues are changed, but the idea remains changeless. The next step in the splinetheory is D.Holladay’s [5] result connecting I.Schoenberg’s cubic spline with thesolution of the problem on minimum of the function norm from the space L(2)

2 .Further, C. de Boor [4] generalized D.Holladay’s result. These results have arousedgreat interest and then appeared a large number of works where depending onthe specific requirements the variational functional was modified. The theoryof splines, based on variational methods, were studied and developed in worksof J.Alberg, E.Nilson and J.Wolsh [1], R.Arcangeli, M.C.Lopez de Silanes andJ.J.Torres [2], M.Attea [3], L.L.Schumaker [6], S.B.Stechkin and Y.N.Subbotin[12], V.A.Vasilenko [13] and others. A fairly complete bibliography in the theoryof spline functions can be found, for example, in [6].

The present paper is devoted to construction of optimal interpolation formulas.Assume we are given the table of the values ϕ(xβ), β = 0, 1, ..., N of functions

ϕ at points xβ ∈ [0, 1]. It is required approximate functions ϕ by another moresimple function Pϕ, i.e.

ϕ(z) ∼= Pϕ(z) =

NXβ=0

Cβ(z)ϕ(xβ)+

2Xα=1

(Aα(z)ϕ(2α−1)(0)+Bα(z)ϕ(2α−1)(1)) (1.1)

which satisfies the following equalities

ϕ(xβ) = Pϕ(xβ), β = 0, 1, ..., N. (1.2)

Here Cβ(z), β = 0, N , Aα(z), Bα(z),α = 1, 2 and xβ (∈ [0, 1]) are the coefficientsand the nodes of the interpolation formula (1.1), respectively.

Optimal interpolation formulas in the space ... 129

Now, following by Sobolev [8], we give the statement of the problem of optimalinterpolation formulas.

We suppose that functions ϕ belong to the Sobolev space L(4)2 (0, 1), where

L(4)2 (0, 1) is the space of functions which are square integrable with 4-th generalized

derivative and equipped with the norm

‚‚‚ϕ|L(4)2 (0, 1)

‚‚‚ =

8<:1Z

0

“ϕ(4)(x)

”2

dx

9=;1/2

(1.3)

and1R0

“ϕ(4)(x)

”2

dx <∞.

The difference ϕ − Pϕ is called the error of the interpolation formula (1.1).The value of this error at a certain point z ∈ [0, 1] is the linear functional on thespace L(4)

2 (0, 1), i.e.

(`, ϕ) =

∞Z−∞

`(x, z)ϕ(x)dx = ϕ(z)− Pϕ(z) = ϕ(z)−NX

β=0

Cβ(z)ϕ(xβ)−

−2X

α=1

(Aα(z)ϕ(2α−1)(0) +Bα(z)ϕ(2α−1)(1)), (1.4)

where δ is Dirac’s delta-function and

`(x, z) = δ (x− z)−NX

β=0

Cβ(z)δ(x− hβ)+

+

2Xα=1

(Aα(z)δ(2α−1)(x) +Bα(z)δ(2α−1)(x− 1)) (1.5)

is the error functional of the interpolation formula (1.1) and belongs to the spaceL

(4)∗2 (0, 1). The space L(4)∗

2 (0, 1) is the conjugate space to the space L(4)2 (0, 1).

Further for simplicity `(x, z) we denote as `(x).By the Cauchy-Schwarz inequality the absolute value of the error (1.4) is

estimated as follows|(`, ϕ)| ≤ ‖ϕ|L(4)

2 ‖ · ‖`|L(4)∗2 ‖,

where ‚‚‚`|L(4)∗2

‚‚‚ = supϕ,‖ϕ‖6=0

|(`, ϕ)|‖ϕ‖ .

Therefore, in order to estimate the error of the interpolation formula (1.1) onfunctions of the space L

(4)2 (0, 1) it is required to find the norm of the error

functional ` in the conjugate space L(4)∗2 (0, 1).

130 Nuraliev F.A.

From here we get

Problem 1 Find the norm of the error functional ` of the interpolation formula(1.1) in the space L(4)∗

2 (0, 1).

It is clear that the norm of the error functional ` depends on the coefficientsCβ(z), β = 0, N , Aα(z), Bα(z) and the nodes xβ . The problem of minimization ofthe quantity ‖`‖ by coefficients Cβ(z), β = 0, N, Aα(z), Bα(z) is the linear problemand by nodes xβ is, in general, nonlinear and complicated problem. We considerthe problem of minimization of the quantity ‖`‖ by coefficients Cβ(z), β = 0, N,Aα(z), Bα(z) when the nodes xβ are fixed.

The coefficients Cβ(z), β = 0, N, Aα(z), Bα(z) (if there exist) satisfying thefollowing equality

‚‚‚˚|L(m)∗2

‚‚‚ = infCβ(z),Aα(z),Bα(z)

‚‚‚`|L(4)∗2

‚‚‚ (1.6)

are called the optimal coefficients and corresponding interpolation formula

Pϕ(z) =

NXβ=0

Cβ(z)ϕ(xβ) +

2Xα=1

(Aα(z)ϕ(2α−1)(0) + Bα(z)ϕ(2α−1)(1))

is called the optimal interpolation formula in the space L(4)2 (0, 1).

Thus, in order to construct the optimal interpolation formula in the spaceL

(4)2 (0, 1) we need to solve the following problem.

Problem 2 Find the coefficients Cβ(z), β = 0, N, A(z), B(z) which satisfyequality (1.6) when the nodes xβ are fixed.

The main aim of the present paper is to construct the optimal interpolationformulas of the form (1) in L(4)

2 (0, 1) space.

Since the error functional ` is defined on the functions of the space L(4)2 (0, 1),

then the following hold

(`, xα) = 0, α = 0, 1, 2, 3. (1.7)

Using the Lagrange method of finding conditional extremum under the conditions(1.7) we get the following system of linear equations for the optimal coefficientsCβ(z), β = 0, N, Aα(z), Bα(z).


NXγ=0

Cγ(z)|hβ − hγ|7

2 · 7!−A1(z)

(hβ)6

2 · 6!+B1(z)

(hβ − 1)6

2 · 6!−A2(z)

(hβ)4

2 · 6!+

+B2(z)(hβ − 1)4

2 · 6!+

2Xα=0

λα(z)(hβ)α =|z − hβ|7

2 · 7!, β = 0, N, (1.8)

NXγ=0

Cγ(z)(hγ)6 + 6B1(z) + 120B2(z)− 2 · 6!λ1(z) = z6, (1.9)

NXγ=0

Cγ(z)(hγ)5 + 5B1(z) + 60B2(z)− 4 · 5!λ1(z)− 4 · 5!λ2(z) = z5, (1.10)

NXγ=0

Cγ(z)(hγ)4 + 4B1(z) + 24B2(z) = z4, (1.11)

NXγ=0

Cγ(z)(hγ)3 + 3B1(z) + 6A2(z) + 6B2(z) = z3, (1.12)

NXγ=0

Cγ(z)(hγ)2 + 2B1(z) = z2, (1.13)

NXβ=0

Cγ(z)(hβ) +A1(z) +B1(z) = z, (1.14)

NXγ=0

Cγ(z) = 1. (1.15)

The solution of this system is the solution of problem 2.The system (1.8)-(1.15) is called the discrete system of Wiener-Hopf type foroptimal coefficients [9, 11]. In the system (1.8)-(1.15) the coefficients Cβ(z), β =0, N , Aα(z), Bα(z) and λα(z), α = 0,m− 1 are unknowns. The system (1.8)-(1.15) has a unique solution and this solution gives minimum to ‖`‖2 under theconditions (1.12)-(1.15) .

Therefore, in fixed values of the nodes xβ the square of the norm of the errorfunctional `, being quadratic function of the coefficients Cβ(z), Aα(z), and Bα(z)has a unique minimum in some concrete value Cβ(z) = Cβ(z), Aα(z) = Aα(z)and Bα(z) = Bα(z).

Here we solve Problem 2 and we find the explicit formulas for coefficientsCβ(z), β = 0, 1, ..., N, Aα(z), Bα(z) of optimal interpolation formulas in the spaceL

(4)2 (0, 1).

The following is true

132 Nuraliev F.A.

Theorem. The coefficients Cβ(z), β = 0, 1, ..., N , Aα(z), Bα(z) of optimalinterpolation formulas of the form (1) in the space L(4)

2 (0, 1) have the followingforms

C0(z) =1

2h7

»− 128z7 + |z − h|7 + (z + h)7 − 14h6A1(z)− 420h4A2(z)+

+10080(λ1(z) + 2λ2(z))h2 +

3Xk=1

Ak

qk

NX

γ=0

qγk |z − hγ|7 +Mk(z) + qN

k Nk(z)

!–,

Cβ(z) =1

2h7

»|z − h(β − 1)|7 − 128|z − hβ|7 + |z − h(β + 1)|7+

+

3Xk=1

Ak

qk

NX

γ=0

q|β−γ|k |z − hγ|7 + qβ

kMk(z) + qN−βk Nk(z)

!–, β = 1, N − 1,

CN (z) =1

2h7

»− 128(1− z)7 + |z − h(N − 1)|7 + (h+ 1− z)7 + 14h6B1(z)+

+420h4B2(z)−10080h2λ1(z)+

3Xk=1

Ak

qk

NX

γ=0

qN−γk |z − hγ|7 + qN

k Mk(z) +Nk(z)

!–,

whereMk(z) =

=

∞Xγ=1

qγk

`(z + hγ)7 − 14(hγ)6A1(z)− 420(hγ)4A2(z) + 10080

`λ1(z) + 2(hγ)2λ2(z)

´´,

Nk(z) =

∞Xγ=1

qγk

`(hγ + 1− z)7 + 14(hγ)6B1(z) + 420(hγ)4B1(z)− 10080(hγ)2λ1(z)

´,

Aα(z), Bα(z), and λα(z) satisfy the following system of linear equations

A1(z)

"−h6

∞Xγ=1

D4(hγ + hβ)γ6

#+ B1(z)

"h6

∞Xγ=1

D4(h(N + γ)− hβ)γ6

#+

A2(z)

"−30h4

∞Xγ=1

D4(hγ + hβ)γ4

#+ B2(z)

"30h4

∞Xγ=1

D4(h(N + γ)− hβ)γ4

#+

+6!λ1(z)

"h2

∞Xγ=1

D4(hγ + hβ)γ2 − h2∞X

γ=1

D4(h(N + γ)− hβ)γ2

#+

+2 · 6!λ2(z)

"h2

∞Xγ=1

D4(hγ + hβ)γ2

#= − 1

14

∞Xγ=−∞

D4(hβ − hγ)|z − hγ|7,


β = −1,−2,−3, N + 1, N + 2, N + 3.Proof. We the system (1.8)-(1.15). We introduce the following denotations

v(hβ) =

NXγ=0

Cγ(z)|hβ − hγ|7

2 · 7!, (1.16)

u(hβ) = v(hβ)−A1(z)(hβ)6

2 · 6!+B1(z)

(hβ − 1)6

2 · 6!−

−A2(z)(hβ)4

2 · 4!+B2(z)

(hβ − 1)4

2 · 4!+

2Xα=0

λα(z)(hβ)α. (1.17)

Then it is necessary to express Cγ(z), γ = 0, 1, ..., N by the function u(hβ).For this we need the discrete analogue D4(hβ) of the differential operator d8

dx8 ,which satisfies the equation

hD4(hβ) ∗ |hβ|7

2 · 7!= δ(hβ),

where δ(hβ) is the discrete delta function. From [7] in the case m = 4 we get thefollowing form of the discrete analogue D4(hβ) of the operator d8

dx8

D4(hβ) =7!

h8

8>>>>>><>>>>>>:

3Pk=1

Ak q|β|−1k for |β| ≥ 2,

1 +3P

k=1

Ak for |β| = 1,

C +3P

k=1

Akqk

for β = 0,

(1.18)

where Ak = (1−qk)9

E7(qk), C = −128, E6(x) = x6 +120x5 +1191x4 +2416x3 +1191x2 +

120x+1 and E7(x) = x7 +247x6 +4293x5 +15619x4 +15619x3 +4293x2 +247x+1are the Euler-Frobenius polynomials of 6 and 7 degrees, respectively, qk, k = 1, 2, 3are roots of the polynomial E6(x), and |qk| < 1 for k = 1, 2, 3.

Using the discrete analogue (1.18) for coefficients Cβ(z), β = 0, 1, ..., N , of theinterpolation formula (1.1) we get the following equality

Cβ(z) = hD4(hβ) ∗ u(hβ). (1.19)

Hence we conclude that if we find the function u(hβ), then the coefficients Cβ(z),β = 0, 1, ..., N, of the formula (1.1) will be found from (1.19).

Now we find explicit representation of the function u(hβ). Since Cβ(z) = 0 forhβ /∈ [0, 1], then from (1.19) we get

Cβ(z) = hD4(hβ) ∗ u(hβ) = 0 при hβ /∈ [0, 1]. (1.20)

134 Nuraliev F.A.

Consider equality (1.16) for hβ /∈ [0, 1].

Suppose β < 0. Then, taking into account (1.9)-(1.15), we have

v(hβ) = − 1

2 · 7!

„(hβ)7 − 7(hβ)6(z −A1(z)−B1(z)) + 21(hβ)5(z2 − 2B1(z))−

−35(hβ)4(z3 − 3B1(z)− 6A2(z)− 6B2(z)) + 35(hβ)3(z4 − 4B1 − 24B2)−−21(hβ)2(z5 − 5B1 − 60B2 + 4 · 5!λ1 + 4 · 5!λ2) +

+7(hβ)(z6 − 6B1 − 120B2 + 2 · 6!λ1)−NX

γ=0

Cγ(z)(hγ)7«. (1.21)

Now we assume β > N using (1.9)-(1.15) from (1.16) we get

v(hβ) =1

2 · 7!

„(hβ)7 − 7(hβ)6(z −A1(z)−B1(z)) + 21(hβ)5(z2 − 2B1(z))−

−35(hβ)4(z3 − 3B1(z)− 6A2(z)− 6B2(z)) + 35(hβ)3(z4 − 4B1 − 24B2)−−21(hβ)2(z5 − 5B1 − 60B2 + 4 · 5!λ1 + 4 · 5!λ2) +

+7(hβ)(z6 − 6B1 − 120B2 + 2 · 6!λ1)−NX

γ=0

Cγ(z)(hγ)7«. (1.22)

Further, using (1.21) and (1.22), from (1.17) we obtain

u(hβ) =

8>>><>>>:(z−hβ)7

2·7! − A1(z)6!

(hβ)6 − A2(z)4!

(hβ)4 + (λ1(z) + 2λ2(z))(hβ)2, β < 0,

|z−hβ|72·7! , 0 ≤ β ≤ N,

− (z−hβ)7

2·7! + B1(z)6!

(1− hβ)6 + B2(z)4!

(1− hβ)4 − λ1(z)(1− hβ)2, β > N.

(1.23)

Here A1(z), A2(z), B1(z), B2(z), λ1(z) and λ2(z) are unknowns.

Now, taking into account (1.23), from (1.20)for A1(z), A2(z), B1(z), B2(z),λ1(z) and λ2(z) we get

D4(hβ) ∗ u(hβ) = 0, β = −1,−2,−3, N + 1, N + 2, N + 3,


that is∞X

γ=1

D4(hγ + hβ)

„(z + hγ)7

2 · 7!− A1(z)

6!(hγ)6 − A2(z)

4!(hγ)4 + (λ1(z) + 2λ2(z))(hγ)2

«

+

NXγ=0

D4(hγ − hβ)|z − hγ|7

2 · 7!

+

∞Xγ=1

D4(h(N + γ)− hβ)

„−(z − 1− hγ)7

2 · 7!+

+B1(z)

6!(hγ)6 +

B2(z)

4!(hγ)4 − λ1(z)(hγ)2

«= 0,

where β = −1,−2,−3, N + 1, N + 2, N + 3.After some simplifications, we get A1(z), A2(z), B1(z), B2(z) λ1(z), and λ2(z)

which are given in the statement of the theorem.Further, from (1.19), using (1.18) and (1.23), for β = 0, 1, 2, ..., N we get

analytic formulas for coefficients Cβ(z), β = 0, N given in the theorem. Theoremis proved.

References

1. J.H.Ahlberg, E.N.Nilson, J.L.Walsh, The theory of splines and theirapplications, Mathematics in Science and Engineering, New York: AcademicPress, 1967.

2. R.Arcangeli, M.C.Lopez de Silanes, J.J.Torrens, Multidimensionalminimizing splines, Kluwer Academic publishers. Boston, 2004, 261 p.

3. M.Attea, Hilbertian kernels and spline functions, Studies in ComputationalMatematics 4, C. Brezinski and L.Wuytack eds, North-Holland, 1992.

4. C. de Boor, Best approximation properties of spline functions of odd degree,J. Math. Mech. 12, (1963), pp.747-749.

5. J.C.Holladay, Smoothest curve approximation, Math. Tables Aids Comput.V.11. (1957) 223-243.

6. L.L.Schumaker, Spline functions: basic theory. Third edition, CambridgeUniversity Press, 2007.

7. Kh.M.Shadimetov. The discrete analogue of the differential operatord2m/dx2m and its construction, Questions of Computations and AppliedMathematics. Tashkent, (1985) 22-35. ArXiv:1001.0556.v1 [math.NA] Jan.2010.

8. S.L.Sobolev, On Interpolation of Functions of n Variables, in: SelectedWorks of S.L.Sobolev, Springer, 2006, pp. 451-456.

136 Nuraliev F.A.

9. S.L.Sobolev, Introduction to the Theory of Cubature Formulas, Nauka,Moscow, 1974, 808 p.

10. S.L.Sobolev, The coefficients of optimal quadrature formulas, in: SelectedWorks of S.L.Sobolev. Springer, 2006, pp.561-566.

11. S.L.Sobolev, V.L.Vaskevich. The Theory of Cubature Formulas. KluwerAcademic Publishers Group, Dordrecht (1997).

12. S.B.Stechkin, Yu.N.Subbotin, Splines in computational mathematics,Nauka, Moscow, 1976, 248 p. (in Russian)

13. V.A.Vasilenko, Spline-functions: Theory, Algorithms, Programs, Nauka,Novosibirsk, 1983, 216 p. (in Russian)

Nuraliev F.A.Institute of Mathematics, Uzbekistan Akademy of Sciences,Mirzo Ulugbek street, 81, 100170, Tashkent, Uzbekistan, e-mail:[email protected]

On a free boundary problem for semilinear parabolic system 137


On a free boundary problem for semilinear parabolicsystem

Rasulov M.S.

Abstract. We study a system of semilinear parabolic reaction–diffusionequations with free boundary. We establish the existence and uniqueness of aglobal classical solution and then study the asymptotic behavior of the freeboundary problem.

Keywords: parabolic system, free boundary problem, a prior bounds,asymptotic behavior of solution

Mathematics Subject Classification (2010): 35K20, 35K51, 35K57,35K58

1 IntroductionConsider the following system for u(t, x), v(t, x) and s(t):

ut − uxx −m1ux = f(u, v), (t, x) ∈ D, (1.1)

vt − dvxx −m2vx = g(u, v), (t, x) ∈ D, (1.2)

u(0, x) = u0(x), v(0, x) = v0(x), 0 ≤ x ≤ s(0) = s0, (1.3)

ux(t, 0) = 0, vx(t, 0) = 0, 0 ≤ t ≤ T, (1.4)

u(t, s(t)) = 0, v(t, s(t)) = 0, 0 ≤ t ≤ T, (1.5)

s (t) = −µ [ux (t, s (t)) + ρvx (t, s (t))] , 0 ≤ t ≤ T, (1.6)

where D = (t, x) : 0 < t ≤ T, 0 < x < s(t), f(u, v) = u(1 − u − kv), g(u, v) =v(1 − v − hu), x = s(t) – moving(free) boundary to be determined with u (t, x),υ (t, x).

Throughout this paper, we will put forward the following assumptions:a. h, k, mi, ρ, µ – are positive constants, i = 1, 2;b. 0 < k, h < 1;c. u0(x), υ0(x) ∈ C2+α[0, s0], u0(x) > 0 Гђ [0, s0], v0(x) > 0 Гђ [0, s0],

u0(0) = v0(0) = 0, u0(s0) = v0(s0) = 0, u′0(s0) < 0, v′0(s0) < 0;

138 Rasulov M.S.

Model (1.1)-(1.6) describes the growth of two populations which are bothdiffusing through D and interacting with each other. The functions representspatial densities and when D is bounded the boundary condition (1.4) imply thatthe populations are confined to D, i.e. there is no migration across the boundaryof D.

For the study of free boundary problems for some biological models, we referto, for instance, [1-4].

The rest of paper is organized as follows. First we establish two-sided boundsfor u(t, x), v(t, x) and s(t), and then a Holder norm bounds for u(t, x), v(t, x).In sections 3, we prove existence and uniqueness results. In section 4, we studyasymtotic behavior of solution (u, v, s) of (1.1)-(1.6) as t→ +∞.

The problem (1.1)-(1.6) was studies in [1-2] for mi = 0, i = 1, 2.

2 A priori estimatesIn this section we establish some a priori estimates of the Schauder type.

Theorem 2.1. Let (u, v, s) be a solution of (1.1)-(1.6). Then

0 < u(t, x) ≤M1, (t, x) ∈ D, (2.1)

0 < υ(t, x) ≤M2, (t, x) ∈ D, (2.2)

0 < s(t) ≤M3 ≡ µ(N1 + ρN2), 0 < t ≤ T, (2.3)

where M1 = max1, ‖u0‖, M2 = max1, ‖v0‖, N1 ≥ maxM1m1, max

x

‖u0‖s0−x

,

N2 ≥ maxM2m2, max

x

‖v0‖s0−x

.

Proof. (f, g) is quasimonotone nonicreasing function in D, as in [5]. Thereforewe will employ the result of the theorem 7.2 ([5], Chapter 8). If we pick ρ1 = M1

and ρ2 = M2 the function (f, g) in (1.1)-(1.2) possesses the property

f(t, x,M1, 0) ≤ 0 ≤ f(t, x, 0,M2),

g(t, x, 0,M2) ≤ 0 ≤ g(t, x,M1, 0).

Then (f, g) satisfies all assumptions of theorem [5]. Thus, we get

0 < u(t, x) ≤M1, 0 < υ(t, x) ≤M2.

Next we claim that 0 < s(t) ≤ M3 for t ∈ (0, T ]. In fact, by the strongmaximum principle and Hopf boundary lemma s(t) is always positive as long


as the solution exists. To drive an upper bound of s(t), we construct auxiliaryfunctions

U(t, x) = u(t, x) +N1(x− s(t)),

V (t, x) = u(t, x) +N2(x− s(t))

and we obtain problem for U(t, x) and V (t, x)8>>>>>>>><>>>>>>>>:

Ut − Uxx −m1Ux ≤M1 −m1N1, (t, x) ∈ D,Vt − dVxx −m2Vx ≤M2 −m2N2, (t, x) ∈ D,Ux(t, 0) = N1, Vx(t, 0) = N2, 0 ≤ t ≤ T,

U(t, s(t)) = V (t, s(t)) = 0, 0 ≤ t ≤ T,

U (0, x) = u0(x) +N1(x− s0), 0 ≤ x ≤ s0,

V (0, x) = v0(x) +N2(x− s0), 0 ≤ x ≤ s0.

If we will choose in the form N1 ≥ maxM1m1, max

x

‖u0‖s0−x

, N2 ≥

maxM2m2, max

x

‖v0‖s0−x

, so that U(t, x) ≤ 0, V (t, x) ≤ 0 holds over D.

Consequently,

ux(t, s(t)) ≥ −N1, vx(t, s(t)) ≥ −N2,

which implies that 0 < s(t) ≤M3, t ∈ (0, T ]. This completes the proof.Using the results of ([6],[7]) we obtain the estimate for | · |D1+α, | · |D2+α.Let

Dδ = (t, x) : 0 < δ ≤ x ≤ s0 − δ, 0 < δ ≤ t ≤ T ,

Dδ0 = (t, x) : 0 ≤ x ≤ s0 − δ, 0 ≤ t ≤ T .

Theorem 2.2. Let M1 = max |u|, M2 = max |v| and assume that u, v, ux, vx

are continuous in D u, v satisfies (1.1)-(1.2) in D together with (1.3)-(1.6). Then|ux(t, x)| ≤ C2(M1,M2), |vx(t, x)| ≤ C2(M1,M2) in Dδ.

Moreover, if the weak second derivatives uxx, utx, vxx, vtx are in L2(D), thenthere exists α = α(M1,M2), such that

|u|Dδ01+γ ≤ C4, |v|

Dδ01+γ ≤ C5.

Additionally, assume that, u, v satisfying (1.1)-(1.2) in D, is continuous with itsux, uxx, vx, vxx derivatives and |u|D2+γ <∞, |v|D2+γ <∞. Then

|u|Dδ

2+γ ≤ C6, |v|Dδ

2+γ ≤ C7.

140 Rasulov M.S.

3 Uniqueness and Existence of solutionFirst we derive an integral expression for the free boundary which will beequivalent to (1.1)-(1.6). Integrating (1.1) over D, we obtain

s(t) = s0 + µ

s0Z0

„u0(ξ) +

ρ

dv0(ξ)

«dξ − µ

s(t)Z0

„u(t, ξ) +

ρ

dv(t, ξ)

«dξ

−µtZ

0

ù(η, 0) +

m2ρ

dv(η, 0)

´dη +

ZZD

`f(u, v)− g(u, v)

´dxdt. (3.1)

Theorem 3.1. Under assumptions of Theorem 2.2, (1.1)-(1.6) has a uniquesolution.

Proof. First, we shall show that uniqueness holds in some smaller value of t.Then, it will be shown that 0 < t <∞. Let s1(t), u1(t, x), v1(t, x) be one solution of(1.1)-(1.6), and let s2(t), u2(t, x), v2(t, x) be another. Let y(t) = min(s1(t), s2(t)),h(t) = max(s1(t), s2(t)). Then each pair satisfies the identity (3.1). Subtracting,we obtain that

|s1(t)− s2(t)| ≤ k1

tZ0

|u1 (η, 0)− u2 (η, 0)| dη + k2

y(t)Z0

|u1 (t, ξ)− u2 (t, ξ)| dξ+

k3

tZ0

|v1 (η, 0)− v2 (η, 0)| dη + k4

y(t)Z0

|v1 (t, ξ)− v2 (t, ξ)| dξ+

+k2

h(t)Zy(t)

|ui (t, ξ)| dξ + k4

h(t)Zy(t)

|vi (t, ξ)| dξ

+

tZ0

dη

y(η)Z0

|f(u1, v1)− f(u2, v2)|dξ +

tZ0

dη

h(η)Zy(η)

|f(ui, vi)|dξ

tZ0

dη

y(η)Z0

|g(u1, v1)− g(u2, v2)|dξ +

tZ0

dη

h(η)Zy(η)

|g(ui, vi)|dξ (3.2)

where ui, vi (i = 1, 2) – is the solution defined between y(t) and h(t), i.e.,

ùi (t, x) , vi (t, x)

´=

(u1 (t, x) , v1 (t, x)), ГҗГҫГҝГї s2 (t) < s1 (t) ,(u2 (t, x) , v2 (t, x)), ГҗГҫГҝГї s2 (t) > s1 (t) .


By the Theorem 2.2, we get

|ui (t, x)| ≤ N1 (y (t)− x) , |vi (t, x)| ≤ N2 (y (t)− x) ,

|u1 (t, y (t))− u2 (t, y (t))| ≤ N1 |s1 (t)− s2 (t)| ,

|v1 (t, y (t))− v2 (t, y (t))| ≤ N2 |s1 (t)− s2 (t)| .Let U(t, x) = u1(t, x)− u2(t, x), V(t, x) = v1(t, x)− v2(t, x). We obtain followingproblem coupled linear parabolic system

8>>>>>>>><>>>>>>>>:

Ut − Uxx −m1Ux − (1− (u1 + u2)− kv2)U = ku1V, (t, x) ∈ D,Vt − dVxx −m2Vx − r (1− (v1 + v2)− hu1)V = hv2U , (t, x) ∈ D,U (0, x) = 0, V (0, x) = 0, 0 ≤ x ≤ s0,Ux(t, 0) = 0, Vx(t, 0) = 0, 0 ≤ t ≤ T,U(t, y(t)) ≤ N1 max

0≤η≤t|s1(η)− s2(η)|, 0 ≤ t ≤ T,

V(t, y(t)) ≤ N2 max0≤η≤t

|s1(η)− s2(η)|, 0 ≤ t ≤ T.

(3.3)

From (3.3), we can conclude that

|U(t, x)| ≤ N1 max0≤η≤t

|s1(η)− s2(η)|+M4 maxD

|V(t, x)|t,

|V(t, x)| ≤ N2 max0≤η≤t

|s1(η)− s2(η)|+M5 maxD

|U(t, x)|t,

where M4 depends on M1 and k, M5 depends on M2 and h.Using arguments and the results [8], we conclude the proof.

Theorem 3.2. Under the assumptions of Theorem 2.1 and 2.2 , there exists asolution u(x, t) ∈ C2+γ

`D´, v(x, t) ∈ C2+γ

`Q´, s(t) ∈ C1+γ ([0, T ]), γ ∈ (0, 1) of

(1.1)-(1.6).

4 Asymptotic behavior of solutionsWe give the main results of this paper.

Let the curve x = s(t) – is increasing function of t and limt→∞

s (t) = s∞ ∈[0,+∞).

Theorem 4.1. If s∞ < +∞, then s∞ ≤ l0 and

limt→+∞

‖u (t, x)‖C[0,s(t)] = 0, limt→+∞

‖v (t, x)‖C[0,s(t)] = 0,

where l0 = minl1, l2, l1 = π√4(1−k)−m2

1, l2 = π

√d√

4(1−h)−m22.

142 Rasulov M.S.

Proof. First prove that, s∞ ≤ l0. Suppose that s∞ > l0, minl1, l2 = l1 andthere exists T > 0 such that l = s (T ) > l1. For this fixed T , we consider followingfunction θ (x) = v (T, x). Then the system (1.1)-(1.6) implies that(

−θ′′ −m1θ′ = θ(1− k − θ), 0 < x < l,

θ′(0) = 0, θ(l) = 0.(4.1)

We have proved that vx(T, x) = θ′(x) < 0, 0 < x ≤ l and θ(x) < 1. If weextend v(x) about x = 0 evenly, then we arrive the following problem(

−θ′′ −m1θ′ = θ(1− k − θ), −l < x < l,

θ(−l) = θ(l) = 0.(4.2)

with θ(x) < 1, −l < x < l. In order to make use of the result of [9], we define thefollowing spectral problem(

−y′′ −m1y′ = λy, −l < x < l,

y(−l) = y(l) = 0.(4.3)

The equation (4.2) has a unique positive solution, if λ1 < 1 − k, where λ1 is thefirst eigenvalue of the problem (4.3).

We can construct λ1 explicitly:

λ1 =` π

2l

´2+`m1

2

´2.

If λ1 < 1− k, then l ≥ π√4(1−k)−m2

1.

Since s∞ < +∞, then s(t) → 0 as t → +∞. Then we can find T < T0, suchthat s(t) < m1

l+s∞s∞

, t ≥ T0. Therefore, as t ≥ T0, x ∈ [0, s(t)], xs(t)l

≤ m1.To compare (u, v) with (u, v) over Ω = (t, x) : t ≥ T0, 0 ≤ x ≤ s(t), it

suffices to show that

ut − uxx −m1ux ≤ u(1− u− kv)

and foru = ν

„l

s (t)x

«, v = 1

we haveut − uxx −m1ux − u(1− u− kv) =

−„s′(t)lx

s2(t)

«ν′ −

„l

s(t)

«2

ν′′ −„m1l

s(t)

«ν′ − ν(1− k − ν) ≤„

l

s(t)

«2 »−ν′′ − s′(t)x+ s(t)m1

lν′–− ν(1− k − ν) ≤


λ1 − ν(1− k − ν) ≤ 0.

We now choose δ ∈ (0, 1) small so that δν (T0, x) ≤ u (T0, x). Then u− (t, x) =

δν (t, x) satisfies8<:ut − duxx −m1ux ≤ u(1− k − u), t ≥ T0, x ∈ [0, s(t)],

ux(t, 0) = 0, u(t, s(t)) = 0, t ≥ T0,u(T0, t) ≤ u(T0, t), 0 ≤ x ≤ s0.

Hence we can apply the comparison principle to conclude that

u(t, x) ≤ u(t, x), t ≥ T0, x ∈ [0, s(t)].

It follows that

ux (t, s (t)) ≤ ux (t, s (t)) = δl

s (t)ν′ (l) → δ

l

s∞ν′ (l) < 0.

On the other hand we have

ux (t, s (t)) = − 1

µs (t) → 0 as t→ +∞.

This contraction prove that s∞ ≤ l1.We now establish that ‖u (t, x)‖C[0,s(t)] → 0 and ‖v (t, x)‖C[0,s(t)] → 0 as

t → +∞. Let l ∈ [s∞, l0] and u(t, x), v(t, x) denote the unique solution of theproblem in the field of Dl = (t, x) : 0 ≤ t ≤ T, 0 ≤ x ≤ l

ut − uxx −m1ux = u(1− u), (t, x) ∈ Dl,

vt − dvxx −m2vx = rv(1− v), (t, x) ∈ Dl,

u(0, x) = u0(x), v(0, x) = v0(x), 0 ≤ x ≤ l

ux(t, 0) = 0, vx(t, 0) = 0, 0 ≤ t ≤ T,

u(t, l) = 0, v(t, l) = 0, 0 ≤ t ≤ T,

where ù0, v0

´(x) =

ù0(x), u0(x)

´0 ≤ x ≤ s0,

0, x ≥ l.

The comparison principle gives 0 ≤ (u, v) ≤ (u, v) in Dl. Then by the property ofthe solution of (5.1) (cf. [9]), we have that u (t, x) → 0, v (t, x) → 0 uniformly inx ∈ [0, l] as t→ +∞.

Therefore limt→+∞

‖u (t, x)‖C[0,l] = limt→+∞

‖v (t, x)‖C[0,l] = 0.

Theorem 4.2. Let (u,v,s) be a solution of (1.1)-(1.6) with s∞ = +∞. Then

(i) limt→+∞

supu(t, x) ≤ 1 and limt→+∞

sup v(t, x) ≤ 1 uniformly in x ∈ [0,∞)

144 Rasulov M.S.

(ii) limt→+∞

inf u(t, x) ≥ 1−k and limt→+∞

inf v(t, x) ≥ 1−h uniformly in any bounded

subset of [0,∞).

Proof. Let u be the solution of ut = u(1 − u) with u(0) = ‖u0 (x)‖C[0,s0].Then it follows that u(t, x) ≥ u(t) for all x ∈ [0, s(t)], t ≥ 0. Taking t→ +∞, weobtain that lim

t→+∞supu(t, x) ≤ 1. Similarly, we have lim

t→+∞sup v(t, x) ≤ 1 and so

part (i) holds.We now prove (ii). For any ε > 0 such that 1− k(1 + ε) > 0, we fix l so that

l > max s0, l0. Since s∞ = ∞ and using (i), one can find tl such that s (tl) = land v(t, x) ≤ 1 + ε for (tl,∞)× (0, l). Let ul solution of8<: ul

t − ulxx −m1u

lx = (ul)

`1− k(1 + ε)− ul

´, t > tl, 0 < x < l,

(ul)x(t, 0) = (ul)(t, l), t > tl,(ul)(tl, l) = (ul)(tl, l), 0 ≤ x ≤ l.

Comparing (ul, 1 + ε) with (u, v) yields that u(t, x) ≥ ul (t, x) in (tl,∞) × (0, l).By theorem 3.3 [9] ul (t, x) → u∗l (x) in C[0, l] as t→ +∞, where u∗l satisfies

−ùl∗´

xx−m1

ùl∗´

x=ùl∗´ `

1− k(1 + ε) + (ul∗)´, −l < x < l,

ul∗ (−l) = ul

∗ (l) = 0.

Thus limt→+∞

inf u(t, x) ≥ ul∗(x) uniformly in [0, l].

On the other hand u∗l (x) → 1− k(1 + ε) uniformly in any compact subset of[0,∞) as t→ +∞ (cf. [10]), which implies lim

t→+∞inf u (t, x) ≥ 1− k as ε→ 0.

Theorem 4.3. Assume that 0 < h, k < 1.

(i) Consider two sequences un, vn defined as follows:

(u1, v1) = (1, 1− h), (un+1, vn+1) =`1− kvn, 1− h(1− kvn)

´.

Then un > un+1 > 0 and vn < vn+1 < 1. Moreover,

(un, vn) →„

1− k

1− hk,

1− h

1− hk

«as n→ +∞.

(ii) Consider two sequences un, vn defined as follows:

(u1, v1) = (1− k, 1), (un+1, vn+1) =`1− k(1− hun), 1− hun

´.

Then un < un+1 < 1 and vn > vn+1 > 0. Moreover

(un, vn) →„

1− k

1− hk,

1− h

1− hk

«as n→ +∞.


References

1. M.Wang, J. Zhao. Free Boundary Problems for a Lotka-VolterraCompetition System. Jour. Dyn.Differ. Equ 26 (2014) 1-21.

2. J. S. Guo, C. H. Wu. On a free boundary problem for a two-species weakcompetition system, J. Dynam. Differential Equations, 24 (2012), 873-895.

3. X.Chen, A.Friedman.A free boundary problem arising in a model of woundhealing. SIAM J.Math.Anal. 32. (2000). 788-800.

4. Y. Du , Zh. Lin. The diffusive competition model with a free boundary:invasion of a superior or inferior competitor. Discrite Contin. Dynam. Syst,B(19), 2014, 3105–3132.

5. C. V. Pao. Nonlinear Parabolic and Elliptic Equations Plenum Press, NewYork, 1992.

6. O. A. Ladyzenskaja, V. A. Solonnikov N. N. Ural’ceva; Linear andQuasilinear Equations of Parabolic Type, Amer. Math. Soc, Providence, RI,1968.

7. S. N. Kruzhkov, Nonlinear parabolic equations in two independent variables.Trans. Moscow Math. Sot. 16 (1967), 355-373.

8. J. Jr. Douglas. A uniqueness theorem for the solution of the Stefan problem.Proc. Amer. Math. Soc. 8. 1952, 402–408.

9. R. S. Cantrell, C. Cosner. Spatial Ecology via Reaction-diffusion Equations.John Wiley and Sons Ltd., Chichester, UK, 2003.

10. Y. Du and L. Ma. Logistic type equations on by a squeezing method involvingboundary blow-up solutions. J. London Math. Soc, 2(64) 2001, 107–124.

Rasulov M.S.,Institute of Mathematics, Uzbekistan Akademy of Sciences,Mirzo Ulugbek street, 81, 100170, Tashkent, Uzbekistan, e-mail:[email protected]

146 Rakhimov D.G., Suyarov Sh.M

Uzbek MathematicalJournal, 2018, No 1, pp.146-??

On eigenvalue problem of the Laplace operators forellipsoidal areas

Rakhimov D.G., Suyarov Sh.M

Abstract. On eigenvalue problem of the Laplace operators for ellipsoidalareas Boundary eigenvalue problems for Laplace operator are solved by regularizedperturbation theory methods in domains with perturbed boundaries.

Keywords: Laplace operator, Banach spaces, eigenelement, eigenvalue,operator-valued function.

Mathematics Subject Classification (2010):

1 IntroductionIn articles [1], [2], perturbations of the discrete spectrum of linear operators actingin Banach spaces using perturbation theory methods and branching theory wereconsidered. In article [3], as an attachment of the results of [1], [2], the problemof oscillation elliptical membrane was considered. In this article, the results of [3]are carried over to a three-dimensional Laplace operator for an ellipsoidal area.In this case, the method of reduction is used, which is developed in article [11].

2 Formulation of the problem

Let E1, E2 be some Banach spaces, A(t) ∈ LE1, E2 be the operator-valuedfunction, which depends on the spectral parameter t ∈ G ⊂ C.

Let ε ∈ C be small parameter, |ε| ≤ %0 and A (t; ε) =∞P

k+l=0

Alkµlεk :

E1 → E2, µ = t − λ0 is the perturbed operator-valued function so thatA (t; 0) = A(t), λ0− is the Fredholm point of the discrete spectrum of theoperator A(t) with N(A(λ0)) = ϕi0n

1 , N∗(A(λ0)) = ψi0n

1 . We need tofind the eigenvalues λ0 + µ(ε) of operator A (λ; ε) , such that µ(ε) → 0where ε → 0, and their appropriate eigenelements. According to the theoremof F.Rellich [1] the operator A (λ; ε) has exactly n eigenvalues λi(ε) (withallowance for multiplicity) with the appropriate eigenfunctions ϕi(ε), ψi(ε), andthe multiplicity of λi(ε) is unknown. Suppose that systems γi0n

1 , zi0n1

biorthogonal to ϕi0n1 , ψi0n

1 respectively.For each i = 1, n we construct the operator

Ai (t) = A (t) +Xj 6=i

〈·, γj0〉 zj0. (2.1)

On eigenvalue problem of the Laplace operators for ... 147

It is simple to verify that λ0 is a Fredholm point for the set of operators(2.1) with the appropriate eigenelement ϕi0 and defective functional ψi0 foreach i = 1, n, i.e dimN

“Ai(λ0)

”= dimN∗

“Ai(λ0)

”= 1. Now if we replace the

operator A(t) by Ai(t) in the expansion of the operator A(t; ε), we obtainthe operators

Ai (λ; ε) ≡ A (λ; ε) +Xj 6=i

〈·, γj0〉 zj0. (2.2)

Definition 2.1. Let us call the condition for the absence of common zeros ofthe operators A(λ0) and

∞Ps=1

As(λ − λ0)s "the condition for the removal of

degeneracy".

Theorem 2.2. Suppose that the condition for the removal of degeneracy issatisfied. If λi(ε) and ϕi(ε), ψi(ε), i = 1, n are eigenvalues and theirappropriate eigenelements and defective functionals of the operator A(t; ε), thenfor each i = 1, n and sufficiently small ε there exist constants cis, dis, s 6= isuch that λi(ε) is also an eigenvalue of the operator (2.2) with the appropriateeigenelement and defective functional

ϕi(ε) = ϕi +Xs 6=i

cisϕs, ψi(ε) = ψi +Xs 6=i

disψs. (2.3)

Proof. If λi(ε) is an eigenvalue of (2.2), then we look for the appropriateeigenelement in the form of (2.3), i.e

0 = Ai (λi; ε)ϕi(ε) =

= A (λi; ε)ϕi +Xj 6=i

cijA (λi; ε)ϕj +Xj 6=i

〈ϕi, γj0〉 zj0 +Xj 6=i

Xs 6=i

cis 〈ϕs, γj0〉 zj0

or0 =

Xj 6=i

cijA (λi; ε)ϕj +Xj 6=i

〈ϕi, γj0〉 zj0 +Xj 6=i

Xs 6=i

cis 〈ϕs, γj0〉 zj0.

After applying functionals ψk0, k 6= i we haveXs 6=i

cis [〈ϕs, γk0〉+ 〈A (λi; ε)ϕs, ψk0〉] = −〈ϕi, γk0〉 , k 6= i. (2.4)

Here A (λi; ε)ϕs = A (λi; ε)ϕi + A (λi; ε) (ϕs − ϕi) = A (λi; ε) (ϕs − ϕi) .Then by the expansions of [1] ϕs(ε) = ϕs0 + εϕs1 + ε2ϕs2 + · · · and A (λi; ε) =A (λ0; 0) +O(ε) we obtain

A (λi; ε) (ϕs − ϕi) = (A (λ0; 0) +O(ε)) (ϕs0 − ϕi0 +O(ε)) = O(ε).

Since 〈ϕi, γk0〉 = 1 + O(ε), the determinant of the system (4) is nonzero, andtherefore it has a unique solution. The uniqueness of ψi(ε) is proved similarly.


Remark 2.3. It is proved similarly that all the eigenvalues of the operators (2.2)are also the eigenvalues of the operator A(t; ε).

3 Perturbation of eigenvaluesFirst we consider the case of the absence of generalized Jordan chains, i.e weassume that 〈A1ϕi0, ψi0〉 6= 0, i = 1, n. Let ϕi(ε) be the eigenelement of theoperator Ai (λi; ε), which is appropriate to the eigenvalue of λi(ε). ThenAi (λi; ε)ϕi(ε) = 0 can be written in the form

A(λ0)ϕi(ε) = A (µi(ε)) ϕi(ε) +Hi (λi(ε); ε) ϕi(ε)−Xj 6=i

〈ϕi(ε), γj0〉 zj0, (3.1)

where A (µi(ε)) = A(λ0) − A (λi(ε)) , Hi (λi(ε); ε) = A (λi(ε)) − A (λi(ε); ε) .After applying the Schmidt’s operator A(λ0) , Γ = A−1(λ0), the equation (3.1)reduces to an equivalent system

A(λ0)ϕi(ε) = A (µi(ε)) ϕi(ε) +Hi (λi(ε); ε) ϕi(ε) + ξizi0,ξi = 〈ϕi(ε), γi0〉.

(3.2)

Writing the first equation of system (6) in the form

ϕi(ε) = ξi

Î − ΓA (µi(ε))− ΓHi (λi(ε); ε)

˜−1ϕi0 (3.3)

and substituting ϕi(ε) into the second equation (6), we obtain the branchingequation for the eigenvalue λ0:

〈Â (µi(ε)) +Hi (λi(ε); ε)

˜ Î − ΓA (µi(ε))− ΓHi (λi(ε); ε)

˜−1ϕi0, ψi0〉 = 0.

(3.4)or taking into account the analyticity of the operator A(t; ε) in the surroundingdomain of the point (λ0, 0), after expansion in a power series and smalloperations

∞Xs=1

∞Xl=0

Lsliµsi ε

l = 0, (3.5)

whereLsli =

X(s,l)=(s1,l1)+···+(sk,lk)

〈As1l1ΓAs2l2 . . .ΓAsklkϕi0, ψi0〉 .

Under our assumptions, the coefficient L10 = 〈A1ϕi0, ψi0〉 is different from zero.Therefore µi(ε) for all i = 1, n are determined from the equation (3.5) in theform of a series in integer exponents of ε. Defining µi(ε) and substituting it inthe formula (3.3), we find the appropriate eigenelement ϕi(ε) in form of a seriesin integer exponents of ε. It means the following theorem is proved


Theorem 3.1. If dimN (A(λ0)) = dimN∗ (A(λ0)) = n and〈A1ϕi0, ψi0〉 6= 0 for all i = 1, n, then for sufficiently small ε there exists exactlyn simple eigenvalues of λi(ε) (λi(0) = λ0) and the appropriate eigenelementsϕi(ε) and defective functionals ψi(ε), analytical by ε.

Now consider the case of the presence of GJCnϕ

(j)i0

oj=1,pi

i=1,n, such that 〈

piPk=1

Akϕ(pi+1−k)i0 , ψi0〉 6= 0 for all i = 1, n. This

means that all GJC should be finite. In this case, in the equation (3.5)

Lpi0 = 〈piP

k=1

Akϕ(pi+1−k)i0 , ψi0〉 6= 0.

The decreasing part of the Newton’s diagram constructed for the branchingequation (3.5) consists either of a line segment joining the points (1, 1) and(pi, 0) (so it will be, if all L0j are equal to zero and L11 6= 0), either fromtwo line segments: of an indicated above and of the segment joining the points(1, 1) and (0, qi) where qi− the number of the first nonzero term in thesequence L0j. The first line segment is responded to the exponent 1

pi−1,

and to the second segment in any case - an integer exponent. Consequently, the

operator A(λ, ε) for sufficiently small ε, has exactly N =nP

i=1

pi (N− root

number) of various continuous by ε eigenvalues λi(ε), λi(0) = λ0, and besidesn eigenvalues are represented by convergent series in integer exponent of ε

and N − n of eigenvalues in powers of ε1

pi−1 . Each λi(ε) corresponds to itseigenelement ϕi(ε) represented by a convergent series of the same exponents ofε, as the appropriate λi(ε). Thus, the following result is proved.

Theorem 3.2. Suppose that in the presence of GJC

Lpi0 = 〈piP

k=1

Akϕ(pi+1−k)i0 , ψi0〉 6= 0 for all i = 1, n. If L0j = 0, j = 1,∞ and

L11 6= 0, then there exists N simple eigenvalues and appropriate eigenelementsthat can be represented in the form series of power of ε

1pi−1 . If L0j = 0, j =

1, qi − 1, L0qi 6= 0, L11 6= 0, then there exists exactly N eigenvalues, n fromthem with the appropriate eigenelements representable over integer exponents ofε, and the remaining N − n with the appropriate eigenelements representablein power of ε

1pi−1 .

Remark 3.3. When L0j = 0, j = 1,∞ and L11 = 0, the methodof the Newton’s diagram allows us to determine all the eigenvalues with theappropriate eigenelements, which can be represented by convergent series infractional exponents of ε by the first nonzero coefficient of the series L1j.


4 The eigenvalue problem of the Laplaceoperator for an ellipsoidal area

Now consider the problem (∆u+ λu = 0, u ∈ Ωε

u = 0, u ∈ ∂Ωε

, (10)

where

Ωε =

(x, y)|x

2

a2+y2

b2+z2

c2≤ 1

ffThe linear transformation ξ = b

ax, η = y, χ = b

cz displays area Ωε into

the ball D =n

(ξ, η)| ξ2

b2+ η2

b2+ χ2

b2≤ 1o. Then equation (10) takes the following

form

b2

a2· ∂

2v

∂ξ2+∂2v

∂η2+b2

c2· ∂

2v

∂χ2+ λv = 0, v|∂D = 0. (11)

Since the ellipsoid has two eccentricities ε1 =q

1− b2

a2 , ε2 =q

1− b2

c2we

consider the special case when a = c 6= b. Then ε1 = ε2 =q

1− b2

a2 . Hence we haveb2

a2 = 1− ε2. Substituting it into (11), we obtain the problem for the perturbationof the eigenvalues:

(A0 − ε2B + λ)v ≡ ∆v − ε2(∂2v

∂ξ2+∂2v

∂χ2) + λv = 0, v|∂D = 0. (12)

We pass to the spherical coordinates in Eq. (12), ξ = ρcosθ · sinϕ, η =ρsinθ · sinϕ, χ = ρcosθ. Then the problem (12) takes the form:

A(ρ, θ, ϕ)w + λv = ε2B(ρ, θ, ϕ)w, w(b, θ, ϕ) = 0, (13)

where B(ρ, θ)w = (1−sin2ϕ ·sin2θ) · ∂2w∂ρ2 + sin2ϕ

ρ2sin2θ· ∂2w

∂ϕ2 + (1−sin2ϕ·cos2θ)

ρ2 · ∂2w∂θ2 −

sin2ϕρ

· ∂2w∂ρ∂ϕ

− sin2θsin2ϕρ

· ∂2w∂ρ∂θ

− ctgθ · sin2ϕρ2 · ∂2w

∂ϕ∂θ+ (1+sin2ϕ·sin2θ)

ρ· ∂w

∂ρ+ sin2ϕ

ρ2sin2θ·

∂w∂ϕ

+ cosθsin2ϕ(1−2sin2θ)

ρ2sinθ· ∂w

∂θ,

A(ρ, θ, ϕ)w = 1ρ2 · ∂

∂ρ

“ρ2 ∂w

∂ρ

”+ 1

ρ2sinθ· ∂

∂θ· sinθ · ∂w

∂θ+ 1

ρ2sin2θ· ∂2w

∂ϕ2 .

It is known that the eigenvalues of the operator A(ρ, θ, ϕ) for the ball are

the numbers λnm =“

µ(n)mb

”2

with the appropriate eigenfunctions

vmnj(ρ, θ, ϕ) =

sπb

2µ(n)m

· 1√ρ· Jn+1/2

µ

(n)m

bρ

!Y (j)

n (θ, ϕ), j = −n, n


where µ(n)m are the roots of the equation Jn+1/2(µ) = 0, Y

(0)n (θ, ϕ) =

P(0)n (cosθ), Y

(−j)n (θ, ϕ) = P

(j)n (cosθ)cosjϕ, Y

(j)n (θ, ϕ) = P

(j)n (cosθ)sinjϕ, j =

1, n. P (j)n (cosθ)− adjointed Lagrange functions:

P(m)n (x) = (1− x2)

m2 · ∂mPn

∂xm.

The problem (10) is self-adjointed, therefore

vjmn(ρ, θ, ϕ) = ψjmn(ρ, θ, ϕ)

zjmn = γjmn =

sπb

2µ(n)m

· 2n+ 1

2πεj· (n− j)!

(n+ j)!· vjmn(ρ, θ, ϕ)

|| 1√ρ· Jn+1/2( µ

(n)mbρ)||2

,

εn =

(2, j = 0,

1, j > 0.

Let λm0n0 =

„µ(n0)m0b

«2

be some fixed eigenvalue of the problem (13). We

perform regularization:

Ai(ρ, θ, ϕ) = A(ρ, θ, ϕ) +Xs 6=i

〈·, γsm0n0〉zsm0n0 ,

then Ai(ρ, θ, ϕ)vm0n0i = 0, i = 1, 2n0 + 1. In accordance with Schmidt’s lemma[7], the operators:

Ãi0(ρ, θ, ϕ)w = A0(ρ, θ, ϕ)w +

2n0+1Xj=1

〈w, γjm0n0〉zjm0n0 ,

where A0(ρ, θ, ϕ)w = ∆w + λm0n0w, are continuously invertible and[ Ãi0(ρ, θ, ϕ)]−1 = Γ for all i = 1, 2n0 + 1. For this we look for a solution of theequation Ãi0(ρ, θ, ϕ)w = h in the form:

w =

∞Xm=1

∞Xn=0

h nXj=−n

Amnj

sπb

2µ(n)m

· 1√ρ· Jn+1/2(

µ(n)m

bρ)Y (j)

n (θ, ϕ)i

Since ∆vimn = −λmnvimn, i = 1, 2n0 + 1, thenA0(ρ, θ, ϕ)vimn = (λm0n0 − λmn)vimn, i = 1, 2n0 + 1. Therefore

∞Xm=1

∞Xn=0

h nXj=−n

Amnj(λm0n0 − λmn)vjmn

i+

∞Xm=1

∞Xn=0

nXj=−n

bZ0

πZ0

2πZ0

Amnjvjmn(r, α, β) ·nX

i=−n

vim0n0(r, α, β)vim0n0(ρ, θ, ϕ)·


hs πb

2µ(n0)m0

2n0 + 1

2πεi· (n0 − i)!

(n0 + i)!· 1

|| 1√ρJn0+1/2( µ

(n0)m0

bρ)||2

i2n0+1

·

r2sinαdrdαdβ = h

Then multiplying both sides by vskl(ρ, θ, ϕ) · ρ2 · sinθ and integrating overr, α, β, we obtain for k 6= m0, l 6= n0, s = −n, n :

(λm0n0 − λkl)Akls ·

s2µ

(n0)m0

πb· 2πεs

2n0 + 1· (n0 + s)!

(n0 − s)!·˛˛

1√ρJn0+1/2(

µ(n0)m0

bρ)˛˛2

=

bZ0

πZ0

2πZ0

h(r, α, β)vskl(r, α, β)r2sinαdrdαdβ

or

Akls =1

λm0n0 − λkl

bR0

πR0

2πR0

h(r, α, β)vskl(r, α, β)r2sinαdrdαdβr2µ

(n0)m0πb

· 2πεs2n0+1

· (n0+s)!(n0−s)!

·˛˛

1√ρJn0+1/2(

µ(n0)m0b

ρ)˛˛2

If k = m0, l = n0

Am0n0s =

bR0

πR0

2πR0

h(r, α, β)vskl(r, α, β)r2sinαdrdαdβr2µ

(n0)m0πb

· 2πεs2n0+1

· (n0+s)!(n0−s)!

·˛˛

1√ρJn0+1/2(

µ(n0)m0b

ρ)˛˛2

We have

Γh = [ Ãi0(ρ, θ, ϕ)]−1h =

bZ0

πZ0

2πZ0

K(ρ, θ, ϕ, r, α, β)h(r, α, β)r2sinαdrdαdβ,

here

K(ρ, θ, ϕ, r, θ′, ϕ′) =

=

nXj=−n

∞Xn=0

∞Xm=1

wmnJn+1/2

“µ(n0)m0

bρ”Jn+1/2

“µ(n0)m0

br”Y (j)

n (θ, ϕ)Y (j)n (α, θ)


where wmn = 1λm0n0−λkl

· πb

2µ(n)m

· 2n0+12πεj

(n0−j)!(n0+j)!

˛˛1√ρJn0+1/2(

µ(n0)m0b

ρ)˛˛2,

wm0n0 = πb

2µ(n)m

· 2n0+12πεj

(n0−j)!(n0+j)!

˛˛1√ρJn0+1/2(

µ(n0)m0b

ρ)˛˛2

Writing for each i = 1, 2n0 + 1 the equation (13) in the form

(Ai(ρ, θ, ϕ) + λ0I)w = −µw + ε2B(ρ, θ, ϕ)w +Xj 6=i

〈·, γjm0n0〉zjm0n0 (14)

we apply the generalized Schmidt’s lemma to equation(14). As a result, we obtainthe branching equations for finding mui(ε):

Fi(µi; ε) =X

k+l≥2

Liklµki ε

l =

= 〈[I + µiΓ− ε2ΓB(ρ, θ)]−1vim0n0 , γim0n0〉 − 1 = 0, i = 1, 2n0 + 1 (15)

The initial coefficients are found as follows:

Li10 = 〈Γvim0n0 , γim0n0〉 = 〈vim0n0 , γim0n0〉 = 1, i = 1, 2n0 + 1

Li02 = 〈B(ρ, θ, ϕ)vim0n0 , γim0n0〉Applying the method of the Newton’s diagram to equations (15), we find:

µi(ε) = Li02ε2 + o(ε2), i = 1, 2n0 + 1,

i.e the eigenvalues of problem (13) have the form of λmni = λm0n0 + Li02ε2 +

o(ε2), i = 1, 2n0 + 1.

References

1. F. Rellich. Storungstheory der Spektalzerlegung. I, Math. Ann., 113, (1936),600-619.

2. T. Kato. Perturbation theory of linear operators., Mir, Moscow, 1972.(Russian)

3. O. P. Bruno, F. Reitich. Boundary-Variation Solution of EigenvalueProblems for elliptic operators. The Journal of Fourier Analysis andApplications, 7, 2(2001), 169-187.

4. B. Simon. Fifty years of eigenvalue perturbation theory. Bulletin of theAmerican Mathematical Society, 24(1991), 303-319.

5. M. R. Dostanic. The properties of the Cauchy transform on a boundeddomain. J. Operator Theory, 36(1996), 233-247.


6. V. A. Trenogin. Perturbation of eigenvalues and eigenelements of linearoperators. USSR Acad. Sci. Doklady. Mathematics,167, 3(1966),519–522.

7. M. M. Weinberg, V. A. Trenogin. The branching theory for solutions ofnonlinear equations. M., Nauka, 1969. (Russian)

8. M.K. Gavurin. About method of false perturbations for the determination ofeigenvalues. J. vich. mat. i mat. fiz., 1, 5(1961), 751-770. (Russian)

9. B. V. Loginov, O. V. Makeeva. Pseudoperturbed iteration methodsgeneralized eigenvalue problems. ROMAI Journal, 1, 1(2008), 149-168.

10. B. V. Loginov, O. V. Makeeva. The pseudoperturbation method ingeneralized eigenvalue problems. Doklady Mathematics, 2009, 77, 2(2009),194-197. Pleiades Publ. Ltd.; Dokl. Akad. Nauk, 2009, 419, 160-163.

11. D.G. Rakhimov. About regularization of multiple eigenvalues by thereduction method of false perturbations. Bulletin of the Samara StateUniversity, Estestvennaya seriya, 2012, є 6(97), page 35-41. (Russian)

12. M. A. Naymark. Normed rings. Nauka, Moscow, 1969, (Russian).

Rakhimov D.G.Lomonosov Mocsow State University (Tashkent branch),100060, Tashkent, Uzbekistan, [email protected] Sh.M.Lomonosov Mocsow State University (Tashkent branch),100060, Tashkent, Uzbekistan, [email protected]



Orthogonality in an abstract spin-factorSeypullaev J.X., Prekeev J.X.

Abstract. In this work we study the connection between algebraicorthogonality, Rober’s orthogonality and orthogonality in the sense of SFS-spacein an abstract spin-factor.

Keywords: Facially symmetric space, exposed face, geometric tripotent,orthogonality, spin factor

Mathematics Subject Classification (2010): 46B20, 46E30

1 IntroductionAn important problem of the theory of operator algebras is a geometriccharacterization of state spaces of operator algebras. In the mid-1980s, Friedmanand Russo wrote the paper [1] related to this problem, in which they introducedfacially symmetric spaces, largely for the purpose of obtaining a geometriccharacterization of the predual spaces of JBW ∗-triples, admitting an algebraicstructure. Many of the properties required in these characterizations are naturalassumptions for state spaces of physical systems. Such spaces are regarded asa geometric model for states of quantum mechanics. In [2], it was proved thatthe preduals of complex von Neumann algebras and, more generally, complexJBW ∗-triples are neutral strongly facially symmetric spaces. In the paper [3]it is proved that the predual space of a real JBW-factor is strongly faciallysymmetric space if and only if it is abelian or spin factor. In the paper [4], thegeometric properties of the cone of positive elements in the abstract spin-factorare studied. The equivalence of the Rober’s orthogonality of elements and theirorthogonality as elements of algebra is established. The present paper is devotedto study of connection between algebraic orthogonality, Rober’s orthogonality andorthogonality in the sense of SFS-spaces in an abstract spin-factor.

2 PreliminariesIn this section we shall recall some basic facts and notation about faciallysymmetric spaces (see for details [1] and [5]).

Let Z be a real or complex normed space. Elements f, g ∈ Z are orthogonal,notation f g, if

‖f ± g‖ = ‖f‖+ ‖g‖.


A norm exposed face of the unit ball Z1 = f ∈ Z : ‖f‖ ≤ 1 of Z is a non-emptyset (necessarilly 6= Z1) of the form

Fu = f ∈ Z1 : u(f) = 1,

where u ∈ Z∗, ‖u‖ = 1. Recall that a face F of a convex set K is a non-emptyconvex subset of K such that if f ∈ F and g, h ∈ K satisfy f = λg + (1− λ)h forsome λ ∈ (0, 1), then g, h ∈ F. An element u ∈ Z∗ is called a projective unit if‖u‖ = 1 and 〈u, F u 〉 = 0. Here, for any subset S, S denotes the set of all elementsorthogonal to each elements of S.

Motivated by measuring processes in quantum mechanics, we defined asymmetric face to be a norm exposed face F in Z1 with the following property:there is a linear isometry SF of Z onto Z, with S2

F = I (we call such symmetries),such that fixed point set of SF is (spFu)⊕ F u (topological direct sum). A real orcomplex normed space Z is said weakly facially symmetric (WFS) if every normexposed face in Z1 is symmetric.

For each symmetric face F we defined contractive projections Pk(F ), k = 0, 1, 2on Z as follows. First, P1(F ) = (I −SF )/2 is the projection on the −1 eigenspaceof SF . Next we define P2(F ) and P0(F ) as the projections of Z onto spF andF , respectively, so that P2(F ) + P0(F ) = (I − SF )/2. A geometric tripotent is aprojective unit u ∈ Z∗ with the property that F := Fu is a symmetric face andS∗Fu = u for some choice of symmetry SF corresponding to F. The projectionsPk(Fu) are called geometric Peirce projections.

By GU and SF denote the collections of geometric tripotents and symmetricfaces respectively, and the map GU 3 u 7→ Fu ∈ SF is a bijection [5, Proposition1.6]. For each geometric tripotent u in the dual of a WFS space Z, we shall denotethe geometric Peirce projections by Pk(u) = Pk(Fu), k = 0, 1, 2. A symmetrycorresponding to the symmetric face Fu will sometimes be denote by Su.

A WFS space Z is strongly facially symmetric (SFS) if for every norm exposedface Fu in Z1 and every y ∈ Z∗ with ‖y‖ = 1 and Fu ⊂ Fy, we have S∗uy = y,where Su denotes a symmetry associated with Fu.

The principal examples of strongly facially symmetric spaces are predualsof complex JBW ∗-triples, in particular, the preduals of complex von Neumannalgebras (see [2]). In these cases, as shown in [2], geometric tripotents correspondto tripotents in a JBW ∗-triple and to partial isometries in a von Neumann algebra.

3 Main resultsElements x and y of Z∗ are orthogonal if one of them belongs to P2(u)∗(Z∗) andthe other to P0(u)∗(Z∗) for some geometric tripotent u.

Proposition 3.1. Let Z be a strongly facially symmetric space and u ∈ GU. Then

P0(u)∗(Z∗) = y ∈ Z∗ : there exist a > 0 with ‖u+ ay‖ = ‖u− ay‖ = 1.


Proof. Let u ∈ GU and

D(u) = y ∈ Z∗ : there exist a > 0 with ‖u+ ay‖ = ‖u− ay‖ = 1.

Suppose that y ∈ P0(u)∗(Z∗), y 6= 0. Then from [1, Lemma 2.1] it follows that

‖u± ay‖ = max(‖u‖, ‖ay‖) = max(1, ‖ay‖),

for all a ∈ R. It follows that ‖u + ay‖ = ‖u − ay‖ = 1, where a = ‖y‖−1. Thisshows that y ∈ D(u).

Let us suppose that y ∈ D(u). Then for any f ∈ Fu we have

|1± af(y)| = |f(u± ay)| ≤ ‖u± ay‖ = 1.

This inequality is valid only for f(y) = 0. Therefore, Fu ⊂ Fu±ay. Then by [1,Lemma 2.8] we obtain that

u± ay = u± P0(u)∗(u± ay) = u± aP0(u)∗y.

Hence, y ∈ P0(u)∗Z∗.Recall that a geometric tripotent u is called maximal if P0(u) = 0.

Corollary 3.2. Let Z be a strongly facially symmetric space and u ∈ GU. Thenthe following assertions are equivalent:

1) u is a maximal geometric tripotent;2) u is an extremal point of the unit ball Z∗.

Let H be a real Hilbert space with a scalar product 〈x, y〉, x, y ∈ H. Considerthe cartesian product A = R ×H = (α, x) : α ∈ R, x ∈ H and define in A theproduct

(α, x) (β, y) = (αβ + 〈x, y〉, αy + βx),

where α, β ∈ R, x, y ∈ H. The norm in A is defined by the formula

‖(α, x)‖ = |α|+ ‖x‖2 (α ∈ R, x ∈ H).

With this product and the norm of sets A is a JBW-factor with unit 1 = (1, 0)which is called the spin factor (see [6]).

Note that in [4] the geometrical properties of the cone of positive elementsin the abstract spin-factor are studied. We establish the equivalence (withsome restrictions) of algebraic orthogonality (that is, when a b = 0) and theorthogonality in the sense of Rober (that is, ‖a + tb‖ = ‖a − tb‖ for each t ≥ 0,the notation of a⊥b) of elements in an abstract spin-factor and is formulated inthe following result.

Theorem 3.3. Let A be a spin-factor. If a = (α, x) ∈ A, b = (β, y) ∈ A andαβ 6= 0, y 6= 0. Then the following statements are equivalent:

1) a b = 0;2) a⊥b.


We note that the implication 1) =⇒ 2) is true for any a, b ∈ A. Moreover, inHilbert space usual orthogonality coincides with the Rober’s orthogonality.

In the paper [3] it is proved that the predual space of a real JBW-factor isstrongly facially symmetric space if and only if it is abelian or spin factor.

The following theorem gives a connection between algebraic orthogonality,Rober’s orthogonality and orthogonality in the sense of SFS-spaces in an abstractspin-factor.

Theorem 3.4. Let A be a spin factor. If a = (α, x) ∈ A, b = (β, y) ∈ A andαβ 6= 0, y 6= 0. Then the following statements are equivalent:

1) a b = 0;2) a⊥b;3) a♦b.

Proof. 3) =⇒ 1). Let a♦b, i.e.

a ∈ P2(u)∗(Z∗), b ∈ P0(u)∗(Z∗),

for some geometric tripotent u. Then by [3, Lemma 1] it suffices to consider thefollowing three possible cases:

Let either u = (±1, 0), or u = (0, z), where ‖z‖2 = 1. In both cases, from theproof of [3, Lemma 1] it follows that F u = 0. Therefore, P0(u)∗(Z∗) = 0, i.e.b = 0. This means that a b = 0.

Let u =`± 1

2, z´, where ‖z‖2 = 1

2. Again from the proof of [3, Lemma 1] it

follows that

Fu =

(1, 2z) : ‖z‖2 =

1

2

ff, F u =

(γ,−2γz) : ‖z‖2 =

1

2

ff.

Therefore,P2(u)∗(Z∗) = span(1, 2z),

P0(u)∗(Z∗) =

(γ,−2γz) : ‖z‖2 =

1

2

ff.

Hence, a = (α, 2αz) and b = (β,−2βz), where ‖z‖2 = 12. Therefore,

a b = (αβ − 4αβ〈z, z〉,−2αβz + 2αβz) = 0.

We note that the implication 3) =⇒ 1) is true for any a, b ∈ A.1) =⇒ 3). Let a b = 0. Then

αβ + 〈x, y〉 = 0, αy + βx = 0.

Therefore, it is sufficient to consider the following three possible cases:Case 1. Let α = 0. Then x⊥y and β = 0 or x = 0. It is clear that if x = 0,

then a = 0 and algebraic orthogonality is equivalent to orthogonality in the senseof SFS-spaces. If x 6= 0, then x⊥y and β = 0.


Suppose that (0, x) (0, y). Since the predual space of the spin factor A is anSFS-space (see [3, Theorem 1]), then from [1, Lemma 2.1] it follows that

‖(0, x+ y)‖ = max‖(0, x)‖, ‖(0, y)‖ = max‖x‖2, ‖y‖2.

On the other hand,‖(0, x+ y)‖ = ‖x+ y‖2.

Hence,‖x+ y‖2 = max‖x‖2, ‖y‖2.

This equality holds if and only if y = 0, i.e. algebraic orthogonality does notimply orthogonality in the sense of SFS-spaces. The case β = 0 is considered in asymmetric way.

Case 2. Let x = 0 (or y = 0). Then α = 0 (i.e. a = 0) or α 6= 0 andsimultaneously β = 0, y = 0. We get the case already considered above.

Case 3. Let αβ 6= 0, y 6= 0. Then x = −αβy and αβ − α

β〈y, y〉 = 0. Therefore,

|β| = ‖y‖ and |α| = ‖x‖.Let α = ‖x‖ and β = ‖y‖. Then

a =

„‖x‖,−‖x‖‖y‖ y

«= ‖x‖

„1,− y

‖y‖

«,

b = (‖y‖, y) = (‖y‖, y) .

Without loss of generality, we can assume that ‖y‖ = ‖x‖ = 12. Then a =

`12,−y

ánd b =

`12, y´. Therefore, by [3, Lemma 1] a and b are geometric tripotents and

‖a± b‖ = 1. Therefore, by [7, Corollary 5] it follows that a b. The case α = −‖x‖and β = −‖y‖ is similar.

Let α = −‖x‖ and β = ‖y‖. Then

a =

„−‖x‖,−−‖x‖‖y‖ y

«= −‖x‖

„1,− y

‖y‖

«,

b = (‖y‖, y) = (‖y‖, y) .

If a b = 0, then (−a) b = 0 and, by the preceding item, (−a) b. It means thata b. The case α = ‖x‖ and β = −‖y‖ is similar.

3) =⇒ 2). Let a♦b. By [1, Lemma 2.1] it follows that

‖a+ (−tb)‖ = max‖a‖, ‖ − tb‖ = max‖a‖, ‖tb‖ = ‖a+ tb‖.

Hence, a⊥b. Note that the implication 3) =⇒ 2) is true for any a, b ∈ A.2) =⇒ 3). Let a = (0, x), b = (β, y). Then a⊥b if and only if x⊥y (see [4]).Suppose that (0, x) (β, y). Then by [1, Lemma 2.1] it follows that

‖a+ b‖ = max‖x‖2, |β|+ ‖y‖2.


On the other hand,‖a+ b‖ = |β|+ ‖x+ y‖2.

Hence,|β|+ ‖x+ y‖2 = max‖x‖2, |β|+ ‖y‖2.

This equality holds if and only if either x = 0, or β = 0 and y = 0, i.e. orthogonalityin the sense of Rober does not entail orthogonality of the sense of SFS-spaces. Thecase β = 0 is considered in a symmetric way.

Let a = (α, 0), (or b = (β, 0)). Then α = 0 (i.e. a = 0) or α 6= 0 andsimultaneously β = 0, y = 0. And we arrive at the case already considered above.

Let a = (α, x), b = (β, y) and αβ 6= 0, y 6= 0. Then the implication 2) =⇒ 3)follows from Theorem 3.3 and the implications of 1) =⇒ 3).

References

1. Friedman Y. and Russo B. A geometric speсtral theorem, Quart. J. Math.Oxford. 1986. N 2 (37). P. 263-277.

2. Friedman Y. and Russo B. Some affine geometric aspects of operatoralgebras, Pac. J. Math. 1989. N 1 (137). P. 123-144.

3. Ibragimov M.M., Kudaybergenov K.K., Seypullaev J.X. Geometriccharacterization of real JBW-factors. Vladikavkaz. Math. Jour., acceptedto publication.

4. Korobova K,V., Khudalov V,T. On the order structure of the abstract spin-factor, Vladikavkaz. mat. journal. 6 (1), 46–57 (2004).

5. Friedman Y. and Russo B. Affine structure of facially symmetric spaces,Math. Proc. Camb. Philos. Soc. 1989. N 1 (106). P.107-124.

6. Ayupov Sh.A. Classification and representation of ordered Jordan algebras,Taskent: Fan, 124 с. (1986).

7. Yadgorov N.Zh., Seypullaev J.X. Geometric properties of the unit ball ofreflexive strongly facially symmetric spaces, Uzb. Math. journal. 2009. N 2.С. 186-194.

Seypullaev J.X.Institute of Mathematics, Uzbekistan Akademy of Sciences,Mirzo Ulugbek street, 81, 100170, Taskent, Uzbekistan, e-mail:[email protected] J.X.Tashkent chemical-technological institute, Navoiy street, 32,100011, Tashkent, Uzbekistan, e-mail: [email protected]

Global solvability to the non-local ... 161


Global solvability to the non-local problem forparabolic equation

Takhirov J.O.

Abstract. In this paper we consider the problem for quasilinear parabolicequation with non-local initial-boundary conditions. A method for establishing apriori estimates of the Schauder type is constructed for solving the problem. Theuniqueness and existence of a solution are proved.

Keywords: parabolic equation, a priori estimates, solvabilityMathematics Subject Classification (2010): 35K57

1 IntroductionQuasilinear parabolic equations and systems of parabolic quasilinear equationsform the basis of mathematical models of diverse phenomena and processesin mechanics, physics, technology, biophysics, biology, ecology, and many otherareas [1, 2, 3]. For example, it arises in mathematical modeling of processes ofchemical kinetics [4], of various biochemical reactions [5], of processes of growthand migration of populations [6, 7], etc.

Such ubiquitous occurrence of quasilinear parabolic equations is to beexplained, first of all, by the fact that they are derived from fundamentalconservation laws (of energy, mass, particle numbers, etc). Therefore it couldhappen that two physical processes having at first sight nothing in common aredescribed by the same nonlinear diffusion equation, differing only by values of aparameter.

In the general case the differences among quasilinear parabolic equations thatform the basis of mathematical models of various phenomena lie in the characterof the dependence of coefficients of the equation (thermal conductivity, diffusivity,strength of body heating sources and sinks) on the quantities that define the stateof the medium, such as temperature, density, magnetic field, etc.

Local (see, for example. [8]) and global ([9]) existence theorems for parabolicequations under different assumptions over the character of non-linearity of theequations have been proved by different methods. Local solvability takes placefor the equations with smooth coefficients without any essential restrictions onthe non-linearity character of the coefficients. Such restrictions appear underconstruction of the global solution.

Problems for linear equations with non-local (initial or boundary) conditionshave been studied in many papers. But boundary value problems for non-linearequations with non-local conditions are almost uninvestigated yet.

162 Takhirov J.O.

In the this paper the problem is considered on the rectangle Q = (t, x) : 0 ≤t ≤ T, |x| ≤ l for the quasi-linear parabolic equation

ut = a(t, x, u, ux)uxx + b(t, x, u, ux), (1.1)

under the conditions

u(0, x) = ku(T, x),−l ≤ x ≤ l, (1.2)

u(t,−l) = u(t, l), ux(t,−l) = ux(t, l), 0 ≤ t ≤ T. (1.3)

It is supposed that the functions a(t, x, u, p) and b(t, x, u, p) are definite for(t, x) ∈ Q and arbitrary (u, p) and bounded on every compactum, 0 < k < 1.

It should be noticed that the when considering the birth process of a separatepopulation in one-dimensional biological reactor having the shape of a long tubeand closed as a ring [10]. In that case u(t, x) is the density of the population, l isthe length of the reactor.

The principal difficulty under constructing the theory of non-local boundaryvalue problems and Caushy’s problem for parabolic equations of form (1.1) is theobtaining of a priori estimations for the absolute value of the derivative ux andits Holder’s constant.

We shall use the following notations from the paper [9]]:

Qδ0 = (t, x) : 0 ≤ t ≤ T, |x| < l − δ,

Qδ = (t, x) : δ ≤ t ≤ T, |x| < l − δ,

Qδ = (t, x) : δ ≤ t ≤ T, |x| < l,

where 0 < δ < min(l, T ).

Some of results [9, 11, 12] will also be used in the investigation of theformulated problem.

Let the function u(t, x) be definite on some set D; for every number γ ∈ (0, 1)let

|u|Dλ = supD|u(t, x)|+ sup

(t,x)∈D,(τ,y)∈D

|u(t, x)| − |u(τ, y)|(|t− τ |+ |x− y|2)γ/2

,

|u|D1+γ = |u|Dγ + |ux|Dγ ,

|u|D2+γ = |u|D1+γ + |uxx|Dγ + |ut|Dγ .

It is supposed that for equation (1.1) throughout the paper the following basicconditions are held:

A. For (t, x) ∈ Q and arbitrary u(t, x) bu(t : x, u, 0) ≤ b0, |b(t, x, 0, 0)| ≤ b1.

B. For (t, x) ∈ Q and arbitrary u(t, x) p(t, x) a(t, x, u, p) ≥ a0 = const > 0.


2 A priori estimations for solving the problemTheorem 2.1. Let the function u(t, x) and its derivative ux be continuous onQ and satisfy the conditions (1.2),(1.3) and the equation (1.1) on Q, may beexcepting points t : 0 ≤ t ≤ T, x = 0. Further, let max

Q|u| = M and for (t, x) ∈

Q, |u| ≤M and arbitrary p the continuous functions a and b satisfy the conditions

a(t,−l, u, p) = a(t, l, u, p), b(t,−l, u, p) = b(t, l, u, p). (2.1)

b(t, x, u, p)

a(t, x, u, p)≤ K(p2 + 1). (2.2)

Then for (t, x) ∈ Q|ux(t, x)| ≤ C(M,a0,K) = M1.

If a1 = max a(t, x, u, p), b2 = max |b(t, x, u, p)| in the domain (t, x) ∈ Q, |u| ≤M, |p| ≤M1, then (see [11], lemma 4.5)

|u(t1, x)− u(t2, x)| ≤ C(M,a1, b2,K,M1)|t1 − t2|1/2. (2.3)

Let u(t, x) has generalized derivatives uxx, utx ∈ L2(Q) then there exists suchγ = γ(M,a0, a1,K); that

|u|Q1+γ ≤ C(M,a0, a1,K), 0 < γ < 1. (2.4)

Proof. The estimation |ux| < C for Qδ directly arises from the results ofthe work ([9], theorem 2). Here the method of introducing of additional spacevariables is used. To complete the proof, it is necessary to establish correctnessof the estimation up to bound of the rectangle Q. In the well-known work [9], asu|x=±l = 0, the author has used the method of continuing the solution through thelateral sides by odd way. But we propose to continue the function u(t, x) throughthe lateral sides of Q according to the rule

u(t, x) = u(t, 2l + x),−3l ≤ x ≤ −l, (2.5)

u(t, x) = u(t, x− 2l), l ≤ x ≤ 3l. (2.6)

It is supposed that the coefficients of the equation (1.1) are continued by xaccordingly (2.5), (2.6). The new function (we shall denote it by u(t, x) too) hascontinuous derivative ux at every point of rectangles R± = (t, x) : 0 ≤ t ≤T, |x± 3

2l| ≤ 3

2l and satisfies "continued“ equation like (1.1) (for example. under

l < x < 3l, ut = a(t, x− 2l, u, ux)uxx + b(t, x− 2l, u, ux)) with the same propertiesas in theorem 1. Using well-known "interior"results, we get an estimation of |ux|in rectangles, which union contains Qδ. As the obtaining of interior estimationsis based on the maximum principle, the statements of the theorem remain true,when the function u(t, x) is continuous in Q, has continuous derivative ux and

164 Takhirov J.O.

satisfies (1.1) throughout in Q excepting points of finite number of straight linesx = const (see the proof of theorem 4.3. in [11]).

Now, using the obtained estimations of |u| in Q and |ux| in Qδ, by virtue of([11],theorem 4.5) we get the estimation (2.3), t ≥ δ.

The interior estimation

|u|Q2δ

1+γ ≤ C(M,a0, a1,K, δ) (2.7)

follows from the results of the work ([9], theorem 3).In order to estimate |uxx(x, t)| in Ω2δ = (t, x) : 0 ≤ t ≤ 2δ, |x| ≤ l and to

find the estimate |u|Q1+γ ≤ C, we will do the following. The problem (l.1)-(1.3) isconsidered in region Ω2δ and using the v(t, x) = u(t, x) − u(0, x) transformation,we have

vt = a(t, x, v, vx)vxx + b(t, x, v, vx),

v(0, x) = 0, (2.8)

v(t,−l) = v(t, l), vx(t,−l) = vx(t, l),

where, the structure of b(·) is well-known. If we prove the holder continuityof uxx(0, x), then the coefficients of the equation (2.8) will satisfy condition oftheorem. After that, applying the well-known result ([8], Theorem 4) to (2.8)problem, we can obtain the estimate |v|Ω2δ

1+λ ≤ C.

In fact, interior result |u|Ω2δ2+γ ≤ C|f |Ωγ . immediately follows from theorem 6

[9]. Next, using the nonlocal condition (1.2), we can obtain |uxx(0, x)|γ ≤ C.

By virtue of the obtained results we can consider the function u(t, x) as asolution of certain linear equation

ut = a(t, x)uxx + b(t, x)

with bounded and continuous, by Holder, coefficients. In order to get theestimation up to the bound, as in the first statement of theorem 1, we shallcontinue u(t, x) by the rule (2.5), (2.6). Further (see the proof of the theorem4 [9]), for the solution of the "interior"equation interior a priori estimations like(2.7) are true in rectangles, containing Q. Under it the results of the work [9] onHolder continuity of the generalized solution are used.

Therefore, we get the estimation (2.4).

Lemma 2.2. For the solution u(t, x) of the problem (1.l)-(1.3) the estimation

|u(t, x)| ≤ b1 expβTβ − b0

= M, (2.9)

is correct. Here β satisfies the condition β − b0 > 0.


Theorem 2.3. In addition to all assumptions of theorem 1, let the inequality|bx| < B1 holds and the function a(t, x, u, p) has continuous bounded derivatives|ax|, |au|, |ap| ≤ K1 the function u(t, x), satisfying the conditions (1.2),(1.3) andthe equation (1.1) in Q, is continuous in Q together with the derivatives ut, ux, uxx,

and on compact subsets of Q - with utx,uxxx. Further, let |u|Qδ

2+γ < +∞ then

|u|Q2δ

2+γ ≤ C(M,a0, a1,K, δ). (2.10)

Moreover, if |u|Q2+γ <∞, then

|u|Q2+γ ≤ C(M1, a0, a1,K1,K). (2.11)

Theorem 2.4. Let the conditions of theorem 2.2 and the concordance conditionb(0,−l, 0, 0) = b(0, l, 0, 0) hold. Then for certain α ∈ (0, 1) there exists the uniquesolution of the problem (1.l)-(1.3) u ∈ C2+α(Q).

Theorem 2.5. Let the function u(t, x) be a solution of the parabolic equation

Lu ≡ a(t, x)uxx + b(t, x)ux + e(t, x)u− ut = f(t, x), (t, x) ∈ Q (2.12)

satisfying the conditions (1.2),(1.3). If the coefficients a, b, e and f(t, x) satisfyHolder’s condition, then there exists solution of the problem (2.12),(1.2),(1.3),such that

|u|Q2+α ≤ C|f |Qα .

The uniqueness of the problem follows from the extremum principle [12].

Remark 2.6. As the uniqueness of the problem is proved under the assumptionof differentiability of the coefficients, then the a priory estimation for higherderivatives has been obtained under the same assumption.

References

1. C.V.Pao, Nonlinear Parabolic and Elliptic Equations. New York: PlenumPress, 1992.

2. R.S.Cantrell, C.Cosner, Spatial Ecology via Reaction-Diffusion Equations.England: Wiley, 2003.

3. A.M.Yaglom, Hydrodynamic Instability and Transition to Turbulence.Springer, 2012.

4. C.Jiang et al., Asymptotic behavior of global solutions for a chemicalreaction model. J.Math.Anal. and Appl. 220 (1998), pp. 640–656.

5. J.C.Arciero et al., Continuum Model of Collective Cell Migration in woundhealing and colony expansion. Biophysical Journal, 2011, v.100, pp. 535–543.

166 Takhirov J.O.

6. J.Lee et al., Pattern formation in prey-taxis systems. J.of BiologicalDynamics, vol.3(6), 2009, pp. 551–573.

7. C.Li, Global existence of classical solutions to the cross-diffusion three-species model with prey-taxis. Computers and Mathematics with Appl. 72(2016), pp.1394–1401.

8. A.Friedman, On quasilinear parabolic equations of the second order II.J.Math. and Mech., vol.9(4), 1960, pp. 539–556.

9. S.N.Kruzhkov, Nonlinear parabolic equations in two independent variables(Russian). Trans. Moscow Math. Soc. 16(1967), pp. 329–346.

10. A.M.Nakhushev, Equations of Mathematical Biology (Russian). Moscow:Vysshay Shkola, 1995.

11. S.N.Kruzhkov, Quasilinear parabolic equations and systems in twoindependent variables (Russian). Tr. of the Seminar im. I.G.Petrovskogo5(1979), pp. 217–272.

12. O.A.Ladyzhenskaya, V.A.Soloinnikov, N.N.Ural’ceva, Linear and QualifierEquations of Parabolic Type (Russian). Moscow: Nauka, 1967.

Takhirov J.O.Institute of Mathematics, Uzbekistan Akademy of Sciences,Mirzo Ulugbek street, 81, 100170, Tashkent, Uzbekistan, e-mail:[email protected]

Mathematical life 167


Mathematical life

Борис Владимирович Логинов

9 января 2018 года после продол-жительной болезни скончался заслу-женный деятель науки РоссийскойФедерации, доктор физ.-мат. наук,профессор кафедры "Высшая мате-матика"УлГТУ Логинов Борис Вла-димирович, внесший огромный вкладв развитие российской и узбекской на-уки.

Логинов Борис Владимирович ро-дился в 1938 году. В 1961 го-ду окончил с отличием механико-математический факультет Ташкент-ского государственного университе-та по кафедре "Математический ана-

лиз". После окончания аспирантуры, в 1965 году - защитилкандидатскую диссертацию "Оценки точности метода возму-щений в теории функциональных уравнений"по специально-сти 01.01.01 - математический анализ. С 1966 по 1993 гг. Бо-рис Владимирович работал в Институте математики им. В.И. Романовского АН Узбекской ССР, сначала старшим науч-ным сотрудником отдела "Дифференциальные уравнения за-тем заведующим отделом "Прикладная математика". В ок-тябре 1982 года в Ученом Совете факультета ВМиК МГУ Б.В. Логинов защитил докторскую диссертацию "Теория ветв-ления решений нелинейных уравнений в условиях группо-вой инвариантности". В январе 1992 г. ему присвоено ученоезвание профессора по специальности 01.01.02 - дифферен-

168 Борис Владимирович Логинов

циальные уравнения. С декабря 1993 работал профессоромкафедры "Высшая математика"Ульяновского государствен-ного технического университета.

Профессор Логинов Б. В. являлся одним из ведущих уче-ных в области нелинейного анализа и нелинейных диффе-ренциальных уравнений. Он разработал новое направлениев области нелинейных явлений - теорию ветвления реше-ний нелинейных уравнений в условиях групповой симмет-рии, получившее международное признание и нашедшее мно-гие приложения в естественнонаучных дисциплинах (поверх-ностные волны, физика фазовых переходов, гидро- и аэро-упругость, нелинейная оптика, математическая биология).

Многолетний цикл работ Б. В. Логинова в области нели-нейных явлений, нелинейного анализа и спектральной теориилинейных операторов отражен в 140 статьях в центральныхи зарубежных журналах, и в 4 монографиях.

Логинов Б. В. - член негосударственных академий и об-ществ: Академия Естествознания (1995 г.), Академия Нели-нейных Наук (1997 г.), GAMM (общество прикладной мате-матики и механики, 1995 г.), AMS (Американское Математи-ческое общество, 1996 г.), ROMAI (Румынское общество при-кладной математики, 2002 г.), член Средне-Волжского ма-тематического общества (Саранск, Мордовия 1999 г.), экс-перт Министерства промышленности, науки и технологий (с2001 г.). Он был членом редколлегий и редактором трудовУлГТУ и УлГПУ, Средне-Волжского математического обще-ства, ROMAI Journal (Румыния), референтом реферативныхжурналов по математике: РЖ "Математика "MathematicalReviews "Zentralblatt for Mathematik". Участвовал на многихВсесоюзных, Всероссийских и международных конференци-ях и симпозиумах.

Под его руководством в период работы в Узбекистане за-щищено 10 кандидатских диссертаций, за время работы в


России.За научные достижения Б.И.Логинов в 1986 г. награжден

орденом "Знак Почета". Имеет отраслевые награды, неодно-кратно поощрялся руководством Института Математики АНУзССР и УлГТУ. 1 февраля 2008 ему было присвоено звание"Заслуженный деятель науки РФ".

Борис Владимирович Логинов до последних дней сохра-нял научную активность, он жил в науке и профессии. Па-мять о нем, как о замечательном ученом и скромным челове-ке, навсегда сохранится и благодарной памяти его ученикови коллег.

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170 Турсун Абдурахимович Азларов


Турсун Абдурахимович Азларов

(1938-2011)

(к 80- летию со дня рождения)

Видному ученому-математику,доктору физико- математическихнаук, профессору, академику, лауре-ату Государственной премии им. АбуРайхана Беруни Турсун Абдурахи-мович Азларову исполнилось бы 80лет.

Т.А.Азларов был достойным уче-ником и достойным продолжателемдела своего учителя, одного из ос-нователей ташкентской вероятностейщколы академика С.Х.Сираждиноваи сделал достойный вклад в развитие

этой школы, воспитал много учеников, оказал большое благо-творное влияние на подготовку высококвалифицированныхнаучных кадров, преподавателей Вузов и школ нашей Рес-публики.

Среди математической общественности в нашей стране иза еҷ пределами Т.А.Азларов широко известен своими глу-бокими исследованиями по ряду направлений теории вероят-ностей и математической статистики, таких как предельныетеоремы теории вероятностей, теория массового обслужива-ния, характеризационные задачи теории распределений, ста-тистические выводы для цепей Маркова и теория приближе-ния непрерывных функций.


Турсун Абдурахимович Азларов родился 15 февраля 1938года в г. Ташкенте. После окончания в 1955 году сред-ней школы (с золотой медалью), он поступает на физико-математический факультет Среднеазиатского Государствен-ного Университета (ныне Национальный Университет Уз-бекистана), окончив его с отличием в 1960 году поступаетв аспирантуру. В 1963 году Т. А. Азларов блестяще защи-щает кандидатскую диссертацию и продолжает работать накафедре теории вероятностей и математической статисти-ки Ташкентского Государственного университета, а затем,в 1970 г. перейдя на работу в Институт математики имениВ.И.Романовского АН Республики Узбекистан, он в 1973 г.защищает докторскую диссертацию и в том году в составегруппы ученых становится лауреатом Государственной пре-мии имени Абу Райхана Беруни.

Работая в Институте математики, сначала старшим на-учным сотрудником, а затем заведующим отделом, ТурсунАбдурахимович успешно сочетает научную работу с педаго-гической деятельностью, работая по совместительству в Та-шГУ (ныне Национальный Университет Узбекистана), снача-ла деканом факультета прикладной математики и механики(1974-76 гг.), а затем профессором кафедры теории вероят-ностей и математической статистики. В этот же период онизбирается членом-корреспондентом (1979г.), а затем и ака-демиком АН Республики Узбекистан (2000г.).

В 1985 г. Т.А.Азларов переходит в ТашГУ, где работаетзаведующим открывшейся по его инициативе кафедрой при-кладной статистики и исследования операций(1985-1992 гг.)и снова деканом факультета ПММ (1988-1992 гг.).

В 1992-1995 гг. Т.А.Азларов - заместитель председателяВАК при Кабинете министров Республики Узбекистан.

С 1995г. по 2006г. Турсун Абдурахимович работает в На-циональном университете Узбекистана заведующим кафед-


рой теории вероятностей и математической статистики и (посовместительству) главным научным сотрудником в Инсти-туте математики АН Республики Узбекистан.

Т.А.Азларов внес большой вклад в развитие теории ве-роятностей, математической статистики и ряда смежных сними направлений математики. Им опубликовано более 160научных работ, в том числе 2 монографии.

Начав под руководством выдающегося математика акаде-мика С.Х.Сираждинова свои исследования в области пре-дельных теорем, Т.А.Азларов уже тогда получил ряд су-щественных результатов, связанных с равномерными оцен-ками в локальных предельных теоремах и эквивалентно-стью локальной и интегральной предельных теорем в некото-рых классах целочисленных случайных величин и векторов.Дальнейшее развитие этих результатов с применениями в ад-дитивных задачах теории чисел, выполненных совместно сего учителем академиком С.Х.Сираждиновым и продолжен-ных Т.М.Зупаровым, Ш.А.Исматуллаевым и другими, послу-жили основой их совместной монографии "Аддитивные за-дачи с растущим числом слагаемых"(1975 г.,Ташкент, изд-во"Фан").

Существенный вклад Т.А.Азларов внес и в развитие ма-тематической теории массового обслуживания. Им доказанасправедливость известной гипотезы Пальма, дано асимпто-тическое разложение для вероятностей потери требования,проведен асимптотический анализ поведения распределенийстационарных характеристик систем массового обслужива-ния, получены равномерные оценки в предельных теоремах,всесторонне изучены переходные явления в системах массо-вого обслуживания с ограничениями, а также в приоритет-ных системах.

Исходя из потребностей теории надежности и массовогообслуживания, им был получен ряд результатов по предель-


ным теоремам для сумм случайного числа слагаемых безпредположения о независимости числа слагаемых от самихслагаемых, включая и равномерные оценки скорости сходи-мости в них, получены асимптотические разложении в пре-дельных теоремах для геометрической суммы случайных ве-личин.

В области математической статистики исследованияТ.А.Азларова, в основном, связаны с систематической разра-боткой теории статистических выводов для однородных це-пей Маркова - классического объекта исследований ташкент-ской вероятностной школы.

Глубокие научные результаты получены Т.А.Азларовыми в исследованиях по характеризации показательного рас-пределения, получивших широкий отклик у специалистов изразных стран и послуживших основой для монографии "Ха-рактеризационные задачи связанные с показательным рас-пределением"(Соавтор Володин Н.А., Ташкент, "Фан 1982г.), переизданной в дальнейшем в Германии издательством"Шпрингер"на английском языке.

К числу нетрадиционных для ташкентской школы отно-сятся исследования Т.А.Азларова по теории приближений.Наиболее общие результаты здесь получены при использова-нии в качестве аппарата приближения класса положитель-ных линейных операторов, определяемых последовательно-стью независимых случайных величин.

Наряду с активной научной работой Т.А.Азларов всегдауделял большое внимание подготовке научных и педагоги-ческих кадров. Им подготовлено 35 кандидатов наук и двадоктора наук. Они работают как у нас в республике, так ив странах ближнего (Россия, Казахстан, Туркмения) и даль-него (Австралия, США, Египет, Монголия) зарубежья.

Т.А.Азларов понимал, что математиков надо готовить сошкольной скамьи принимал активное участие в проведении

ежегодных математических олимпиад (начиная со школьно-го и кончая всесоюзным уровнями). Совместно с другом иколлегой проф. Мансуровым Х.Т. им был написан ориги-нальный учебник по математическому анализу на узбекскомязыке (в двух томах), выдержавший не одно издание (в част-ности и его переработанный вариант на латинице). Под егоруководством и при его непосредственном участии изданофундаментальное пособие по самообразованию для учителейматематики на узбекском языке.

За большие заслуги в научной и педагогической деятель-ности Т.А.Азларов был награжден орденом Знак Почета имедалями.

Т.А.Азларов был активным участником многих междуна-родных научных форумов. Он принимал участие в тради-ционных Ферганских и Вильнюсских конференциях, I Все-мирном конгрессе общества Бернулли и других международ-ных симпозиумах, причем во всех из них, проводившихся вУзбекистане, он выступал также и в роли активного орга-низатора. Т.А.Азларов неоднократно выезжал в зарубежныестраны для чтения лекций, научной работы или участия вконференциях (Япония, Индия, Венгрия, Польша, Финлян-дия, Швеция, Египет и др.).

Вся многогранная жизнь замечательного ученого с высо-кими человеческими качествами является примером безза-ветного служения науке, образованию, народу.

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Компьютерная верстка: K.K. Abdurasulov

Журнал зарегистрирован Агентством по печати и информацииРеспублики Узбекистан 22 декабря 2006 г. Регистр. 0044.

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