On Double Sturm-Liouville Series

On Double Sturm-Liouville SeriesAuthor(s): Josephine MitchellSource: American Journal of Mathematics, Vol. 65, No. 4 (Oct., 1943), pp. 616-636Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371870 .

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ON DOUBLE STURM-LIOUVILLE SERIES.*

By J'OSEPHINE MITCHELL.

Introduction. In this paper we consider the double Sturm-Liouville series (SL series) of a functioni, Lebesgue integrable 1 on the square Q (0 < x c 7r,

O ? < 7r)y that is, the Fourier development of this function with respect to the double orthonormal system, {fpm (x) ckn (y) }, (n, n =- 0, 1, 2, ) (1). Our main method is the comparison of the double SL series of a given fullc- tioln f (x, y) with its Fourier cosinie-cosine series (FCC series). A. Haar [91 introduced this idea in the one-dimensiolnal case, proving that the differenice of the partial sums of the SL and Fourier cosine developments of ail integrable function f(x) converges to 0 uniformly on (0, 7r) ; briefly, the two series are said to be uniforlmlly equicon'Verqge)nt.2 The proof depends mailylr upon the fact that the difference between the SL aiid cosine kernels (the differeince kernel) is boundled [9].

It is natural to generalize the concepts of " equiconxvergence " and " equisummability " to the double SL andl FCC series. However the " cross product" terms of the correspondinig clifference kernels are not bounded (as Haar has pointed out), so that in Lhe case of equiconvergence we have fouilid it necessary to put some restrictionl on the given functionl to be expandecl.

Ill discussing the equiconvergence of the double SL and FCC series (Part ITI) we assume that the givei function is of bounded variatioln (Tonelli) ([17], p. 443) and obtain equticonvergence of the SL and FCC' series over the square Q (Theorem III). Since this conditioll is included in all of the older sufficient conditions for the convergence of the double Fourier series as well as in some of the more modernl ones,3 we get various sufficient conditions for the convergence of the double SL s-eries (Theorem IV).

" Received July 1, 1942; Revised January 18, 1943. In this paper integrationi is Lebesgue integration. Also, a measulable function

f is said to be of class LP if f IP is initegrable. 2 We may consider instead of convergence somiie of the various methods of suminia

tion of series anid define equisinnrmability of two series. 3Well-kniown sufficient conditions for the conver-enice of the double Fourier series

are, e. g., that the functioni be of bounided variation (Hardy) [10] or of bounded varia- tionl (Arzela) [7]. Other sufficient co,nditionis are given by L. Tonelli [17] anid in g,eneralizations to two variables of the tests of Dini, de la Vallee Poussin, Lebesotue, etc. For a statemeniit of the tests and the conniectioiis among them see J. J. Gergenis paper [8].

616



ON DOUBLE STURM-LIOUVILLE SERIES. 617

Part II contains our prinicipal theorem (Theorem I) on the (C, 1, 1) equisummability of the double SL and IF'CC series, namely that the SL series of an integrable function f (x, y) is (C, 1, 1) equisummable with its FCC series at every point (x, y) in Q for which

l r h ek (A) 7Ij Jff(x? uy,Y v),dudv? C(x,y) < oo, ( < I h <7rO < Il k!? )

We also prove that the two series are equisummable almost everywhere (a. e.) in Q (Theorem II).

To get a summability theorem for double SL series we consider the corresponding theory for the summability of the double FCC series. In 1918 H. Geiringer ([7], also [17]) following the method of proof introduced by Lebesgue, proved that the Fourier series of f(x, y) is (C, 1, 1) summable to f(x, y) at every point (x,y) where

1 h rk (C) lim Ik J f f(x+u, y+v)-f(, y)I dutdv=0o

h->o, ->o hc o

if, also, the two simple integrals of I f I are bounded. A. Zygmund [19] proved the summability at all points (x, y) for which (C) holds and

(D) I fxh r u f( y vy+v) -f(x y)I dudv < D(x, ) < tC,

(O < I h I _ 7r, 0 < 71 )

Relating these conditions to the problem of "strong derivability" of double integrals, which has been thoroughly investigated in the last few years by S. -Saks, A. Zygmund and others [19], he showed that (C) and (D) hold a. e. for a function of class LP (p > 1) and that therefore the Fourier series of such a function is summable a. e. A more recent investigation by B. Jessen, J. Marcinkiewicz and A. Zygmund [16] has extended this result to a class of functions f (x, y) such that f log+ I f I (cf. footnote 11) is integrable (denoted by class (B)). By showing that for such functions condition (A) holds a. e.4 we deduce that the SL series of a function f (x, y) of class (B) is (C, 1, 1) summable to f (x, y) a. e.

Finally in part IV we consider a modification of the Abel-Poisson summation method. Instead of multiplying the n-th term of the Fourier develop-

00

ment, E cnvn(O), of a given function f(O), defined on (- r, 7r), by rn, n=O0

we form the sequence {v.(G, r) }, (n = 0, 1, 2 ), where vn(0, r) is har-

4For a discussion of the relation between functions of class (B) and conditions

(C), (D) and (A) see 4.

7



618 JOSEPHINE MITCHELL.

monic in the inzterior of the unit circle and reduces to vn(O) on the circumt- 00

ference and consider the limit of the series , cGvt(6, r) as r approaches 1 -0 n=O

(modified Poisson method of summahility) (6. 1). For ordinary Fourier series this reduces to the usual Abel-Poissoni summability. This procedure, which can be generalized in various ways to more variables, was introduced by S. Bergman ([4], [51) in a much more general connection in the theory of functions of two complex variables. Applying this summation to the SL system we show that the double SL series of an integrable function is equisummable with its FCC series (Theorem V). Lastly we state a theorem on modified Poisson summability for the development of an integrable function with respect to more general orthonormal systems (Theorem VI).

PART IL The Sturm-Liouville System

1. Definitions and theorems.

1. The SL system ([9], [181). Consider the SL orthonormal system {+n(X)}, (n=O, 1, 2, )- defined on (O??xir). The (An(x) are the characteristic functions of the differential system

(1) d u/dx2+ (Q((x) +A)u=0

du/dx -hu 0 for x-O, du/dx + Hu O0 for x 7r(h,H conistants),

where X is a parameter (- oo <A < oo) and Q(x) is continuous. If, also, (Q(x) is of bounded variation, Liouville, and Hobson [12], have proved the asymptotic formula

(2) On(x) = (2/sr) cosnx + s(x)in n( n (0) (= 1, 2 (2/w)?conx?p(x) n +(n?+1)2'

where the function /38(x) has a continuous derivative, the continuous functions pn(x) are bounded in absolute value independently of n and x, and for n = 0 the second term on the right is zero.

Now let f(x) be a function, integrable on (0, 7r). The development of f (x) by the SL system

00 7r

(3) E anOn (X), an f (s) ot, (s) ds, (n = O,~ 1. 2, *)

is called the SL series of f (x). It bas been provevd that the SL system is complete with respect to in-

tegrable functions [18].



ON D)OUBLE STURM-LIOUVILLE SERIES. 619

Using the system {q,n(x) }, we can form the double SL orthonormal system {Jm(x)+0n(y)}, (m,nO0,1,2, ), where (x,y) is a point in

Q(O?x?7r, 0 y-- ). Now let f(x, y) be an integrable function in Q.5 Its double SL series is

00

(4) i a,nu4rn (X) >n (Y) mn=0, n=o

where <7r r

(5)) amn= f (sf t)f( t> m(S)n(t)ds dt, (m,n ?1 2, .

In this paper we compare the double SL series of f (x, y) with its double F5CC series

00

(6) E Xmnbiin> cos mx cos ny in=O, n=O

vt?mn if m r n -O. 1 if n =O, n > O or m > O, nO, 1 if M > O, n > O where

4 , Jr , (7) bjn 7 f (s, t)cos rns cos nt dsdt, (in, n 0, 1, 2,

2. Equiconvergence and (C, 1) equisummability for f(x).

Denote the partial sums of the SL atnd cosine kernels, respectively, by

n (8) Ktr (X, S) E OJV (x)oV (s)

P=o and

(9) ~(71 (x, S) (2/7r) (+ cos vx cos vs), 1,=j

and define the "difference kernel," 4)n(X, s), by

(1 0) @ DI (aX S) =.K1" (X ,s) -C (X,s ).

Similarly denote the correspolldiDg acrithmnetic means ((C, 1) means) by

(<11) t'"K*n(X, 9) Ko(x, s) * +K.,,(x, s)

and

(12) (i*,b(X,s) Co(x,s ) +*- + C, U1(x,s) n

and define the " (C, 1) difference lkernel," Vn *(X, s), as in (10).

Whenever we consider (x, y) beyond the square Q, we extend it as an even-even function of period 27r in each of the variables x and y.

6 The completeness of the double SL system with respect to integrable functions is an immediate consequence of the corresponding property for the simple system ( [20] p. ]3, Ex. 6).



620 J OSEPHINE MlITCHELL.

Haar's proof of the equiconvergence theorem and the analogous proof for the equisummability theorem involve the boundedness of the difference kernel [9], that is,

(13) bn (X7 S) | @

where 1 is independent of n, x and s, and

(14) ! @*n(X,s) I I 'o (X, s) + * *?'n-, (X s) Ic

THEOREM 1 7 (Haar). The SL series of a function f(x), integrable on

(0, 7r), is uniformly equiconvergent with its Fourier cosine (EC) series [9].

THEOREM .2. The SL series of a, function f (x), integrable on (0, 7), is uniformly equisumnmable (C, 1) with its FC series.

2. Two properties of the difference kernels ,(x, s), V?*,(x, s). The following lemmas enable us to prove general (C, 1, 1) equisummability theorems for the double SL and FCC series , (cf. Part II).

LEMMA I. T7he difference kernel 4n(x,s) converges to 0 uniformly with respect to s as n o-> o if s is in any closed interval not containing the point x.

Proof. Let x be an arbitrary fixed point in (0, 7r) .

From the asymptotic formula 1 (2) for the SL functions we easily see that the sequence cDn (X, S) converges uniformly for s in any closed interval (a, b) in (0, 7r), not containing the point x, as n -4 oo. Hence the limit

(D (x, s) is a continuous function of s in (a, b). Further I Pn(x, s) ! _ so that

Li '

)lim 'bn (x, s) ds tf(x, s)ds. n->oo a

But from llaar's theorem

s4

(2) lim Dn(x, s)ds-O, ,n--ooa

for fK (x, s) ds is the partial sum of the SL development of the function

equal to 1 in (a, b) and 0 elsewhere, and similarly for C3 (x, s) ds.

Consequently 4F(x, s) = 0 for all s in (0, 7r) different from x.

7 We indicate known theorems and lemmas by arabic numbers and new ones Roman numerals.

8We intend to uise Lemmas I and II agaiin in a later discussion on the eqluic vergence of the SL and FCC series.




This same property is, of course, valid for the kernel ' (x, s) defined in 1.9

LEMIMA II. 7r

(3) lim fi. (x, s)lds =0 for all x in (0,). n->oo 0

Proof. This property follows for each x in (0, ir) from Lemma I by the Lebesgue theorem on the integration of sequences. The same property holds for the kernel 48*n (X, s).

COROLLARY. If f (x) is integrable over (0, 7r), then for all x in (0, 7r)

~7 (4) lin (x, s) f (s) ds = 0.

5n-> o

PART I. On the (C, 1, 1) Equisummability of the double SL and double FCC Series.

3. Theorem on Equisummability.

1. THEOREM I. Let f(x, y) be integrable in the square Q (O ? xa;< ?r, 0 ? y ? 7r). Then the double SL series of f (x, y) is (C, 1, 1) equisummable with its double FCC series at every point (x, y) in Q for which

(A) hk j f f(x + i, y+ v)I du?dv?C, (O <jhI!7,0<Ikl?7r)

where C =C(x,y) < c<o10

Proof. Denote the SL and FCC (C, 1, 1) partial sums, respectively, by r7r4

( 1 ) omn (x,yG = ffK Km (x, s) K*n (y, t) f (s, t) dsdt

(2) a m(n y(X,Y) = Cm (x, s) Gn (y t) f (s, t) dsdt.

Then a.n,, (x, y) -0,nn (X, y) equals the sum of the three integrals

(3) 41(a, y) ? *,, /(X, s) *n (y, t) f (s, t)ds dt,

(4) nn (X y) J 5 5C k( s) * n (y, t) f (s, t) dsdt,

( 5 ) (3) (Xay) x = f 5 m(aS) C n(y, t) f (s, t) dsdt.

'It is well-known that Lemma I is valid for the Fejer kernel CG (x, s). Thence it also holds for the SL kernel K*n (X, s). However we do not use this fact.

10 FoI a further discussion of conditioni (A) see 4. 2.




It will, obviously, be sulicielit to prove that I 0 |, |(2) I and j (3 0 mn "in ~mn

as n, it- o.

2. The fact that |i'1)(x, y)| 0 follows at once from the uniform- boundednesss of cD*m

Now 1 sin2 jM(s x) sin2 1m(s + x)

(6) C*m (X, S) =2 mw ( 2sin2 (S-_X) +2Sin2 (S +X)

so that in discussing 4I(2) (x; y) we consider the two integrals "in

(7 ) i'n(xy) = nJn f sin2 im YS+ ) *2 (y, t) f (s, t) dsdt

and

(8) nn(X, Y) = Lf7rI f i2im(s $) D*n(yt)f(s, t)dsdt.

If x = 0 or 7r, tI"n (x, y) reduces to the same value as '".mn (x, y). If x is an interior point, we can choose a 8 > 0 so that 0 x - 8 < x + _<7r

and then

(9) Vni ' n(X, Y) -n 12 bt C if(s, t) I dsdt. mw7 2sin I~ Jo .I

For fixed S, a sufficiently large m gives

(10) V 'mn(X, y) j < rj, where r, is an arbit-rary positive number.

To consider the integral '', (x, y) we decompose it into

(ii) f f= X f+ H 4 J~ f2 L' 31} o o o v Z o . U-6 y V+e e

Let

(12) g(h,1)= If(x+h,y+v)ldv,6(h. k)= Jf f(x+u, y+ v)Idudi.

It is readily shown that

(13) ? sin2 - = < for 0 < u?= 7r (e. g. [17]. p. 176). m.' Sin 2 ~u (+ MU)2 Then

1 ~~~~'7 j2 1M ( X

( 14) 1~j ma(r , 2si2m(s-) I,' y, t) f(s, t) dsdt I

1 fr-X fSin2 1rM c Jo o 2 X sin32 iu ! f(x + u, y + v) Idudv

??4 {1 + I MU




Tltegration by parts of the right band integral with condition (A) gives us

(15) MO (, m (u)7r) +m S)

- 2b ,( I+ 2lM7r)2 + l (1 + 1m ,3 d __ 5 ~MCwE 7 m 2u

(1 + im7r)2+2E }

Secondly let

(16) max 1 3I4)= (y, t). V+ec?t?r

By Lemma I, AJ,, -- 0 as n oo for fixed e. Then

(17) 2 ! m7r JwJy+e 2sin2 1 (s-x) '*(y,t)f(s,t)dsdt

<17rl. ! 1? .

g(u,7r) du.

Again integration by parts with condition (A) gives

(18) I2 1?73.2 { (l?+ 2+2 }

Now it is readlily seen that 4'mm decomposed as in (11), involves four integrals of type I. and four of type '2, satisfying inequalities like (15) and (18) respectively. From inequalities (15) and (18) and Lemma 1, it now follows at once that limJ"', (x,y) =0. Consequently

m,n

(19) lim n (V., y) 0. m-oo, n>cc,

Similarly for !t') (x, y) I and the theorem is proved.

4. Summability for the double SL series and further results on Equi- summability.

1. It is easily deduced that condition (A) is satisfied at almost every point (x, y) provided that

(B) f(x,y)log+If(x,y), belongs to L over Q 1 ([14], p. 221).

On the other hand it is well-kinowin that the Fourier series of functions satisfying condition (B) are summable (C, 1, 1) to f(x, y) almost everywhere [14].12 It follows that the double SL series of the function f satisfying (B) is summable (C, 1, 1) to f(x, y) a. e.

11 Log+ I f (x, y) I is defined as equal to log I f (x, y)! if I f (x, y) I > 1, equal to 0 otherwise (cf. e. g. [20], p. 150). f log+ I f I of class L implies f of class L ([11], p. 99).

12 If f (x, y) is extended beyond Q as an even-even function its double Fourier series reduces to the douible FCC series.



624 JOSEPHINE MITaCTIELL.

We shall prove now the following

THEOREM II. Let f(x,y) be integrable over the square Q. Then the double SL series of f(x, y) is (C, 1, 1) equisum,mable with its double FCC series at almost every point (x, y) in Q.13

Proof. It is sufficient to prove that *J(2) (x, y) > 0 a. e. We use the fact

that 4 C*, (x, s) f(s, t) dsdt converges to f (x, t) dt at all points x wh >7r

where the derivative of I f (x + u, t) -f (x, t) I dudt exists and equals

0 (a. e.). 14

Now (1) !I,(2)(X,Yy)I?Jl+ IJ2 wvhere

(2)) PJ1 -C*ffn (x( s)Xb)*n(y, t)f(x, t)dsdt

7r - f *tt (y, t) f (x, t) dt

7r > r

() J2 m (xr(s) 4)b*n (yn t) (f (s, t) -f (x, t) )dsdt.

By Fubini's theorem f(x, t) is integrable with respect to t for a. a. x. There- fore by Theorem 2 the right side of (2) > 0 as n-> oo for a. a. x and uni- -formly in y. Hence JJ1-O0 as n. oo for a. a. (x, y) in Q.

Again 7r

(4) J2 C*m (x, s) g (s) ds

wrhere g (s) f (s, t) - f(x, t) I dt. From the theory for simple Fourier

series the second factor on the right of (4) -> 0, as m -> oo, at all points x for which the derivative of the indefinite integral of g (s) is 0 (a. a. x). Hence J-> 0 as m -*> for a. a. (x, y) in Q. Consequently I (2) -> 0 as m. n -> oc. Similarly for I f(3) and the theorem is proved.1

2. It has been proved that the double FCC series of an integrable function f(x, y) is (C, 1, I) summable to f(x, y) at all points for which the two

" This theorem was suggested by the referee. 11 The latter statement can be proved by a generalization of the proof of the

corresponding one-variable theorem given in ([13], vol. 1, p. 58,2). 15 It must be noted that this proof of the fact that I 4 (2) 1 -e 0 does not use

ma Lem-ma I. However the proof seems to us less straight-forward than that of Theorem I.




conditions (C) and (D) (see Introduction) hold and that for functions of class LP (p > i) these conditions hold a. e. [19]. We now point out that eveni for functions of class (B) condition (C) holds a. e. and that moreover (C) implies (T)) a. e., thus affording a second proof of the summability theorem for suich functions; since obviously (D) implies (A) everywhere we have a proof of the corresponding summability theorem for double SL series which shows the connection of these different conditions.

This fact follows for condition (C) from the recently proved theorenm that the integral of a function f of class (B) is "strongly differentiable" 16

to f(x, y) a. e.7 by Lebesgue's argument ([13] vol. 1, p. 582).

Condition (D) can be proved to result from (C) a. e. as follows. If h, k are sufficiently small (< ki) (D) is given by the statement of condition (C). Also (D) follows readily if h, k ? 81. If h < 8, and ki, > 8,, then

I h -a

(5) f f f(x u,+yu +v))-f(x,Y) Idudv

Ia,f (C I f (x +%t) I dt) du +,If(x, y

Now the monotone function (f f(s, t)I dt) ds is differentiable to

fI f (x, t) 1 dt at a. e. point x in (O, 7r) (even if f (s, t) belongs to L). Con- sequently for such x and sufficiently small h (h < 82)

Hence

( ) hk Sot 7o 1 f (x + %y L- v) f (x,y) I dit dv

C 8fff uI v) (Co t)xIdt +)fd(xud)

A similar result holds if h ?. .8, and Ic < 8,. Therefore from the results of

the four cases condition (D) fol]ow-s from (C) a. e.

Stro'ng der ivability of a dlouble integ,ral mieans that

(1/hk) t (x ?+it, , + ?v) dudv f(x, y) as h, k---0.

17 Such theorems have been proved by S. Saks, B. Jessen, J. Marcinkiewicz and A. Zygmund, etc. For the particular statement of the theorem used here see ([16],

p. 122).




PART III. On the uniform Equiconvergence of the double SL and FCC Series.

5. Equiconvergence theorenm and sufficient conditions for the, convergence of the double SL series.

1. In this section we prove a theorem on the equiconivergence of the double SL and FCC series of an integrable function f(x, y). Since the difference kernel is not bounded a proof similar to that of Haar's theorem cannot be generalized to two variables. Further, the Fourier kernel is not positive as in the case for Ces'aro sumrmability. We prove equiconvergelcee for integrable functions f(x,y)' which are of bounded variationt as definecl )y Tone7li, (b.v. T.). This condition is only slightly stronger than that of

bounded -variation in eaeh variable and is by no means sufficient for the conivergence of the double Fourier series as L. Tonelli has shown by a counter- example ([17], p. 4805).

DEFINITION. -A function f(sx, y) is said to be of bounded variationi (To relli) in Q if

(i) f(x, y) is of bounded variation in (0, 7r) as a function of y for allmost all x,

(ii) f(x, y) is of bounded variation. in (0, w) as a function of x for a. a. y,

(iii) the total variation of f(x, y) with respect to y, namely,

(1) VI(z) SUP E I f(X, gv) -f(x, yV1) l,C; 0O-yo < Y < ..<yk 7r< is dominated by an integrable. function UL(x); that is,

(2) V17(x) U(x),

(iv) and similarly the total variaction of f (x, y) with respect to x, namely? I2(y), is dominiated by an integrable functiom- U2(y) ([17], p. 443).18

2. THEOREM III. If the integrable function f(x, y) is of b. v. T. itn square Q, then its double SL series is equticonvergent in Q with its double FCC series.

Proof.19 In analogy to the notation of 3. 1 call the partial sums of the double SL and FCC series

18 L. Tonelli imposed originally integrability on V1 (w) and V2 (y). The extension considered here is due to C. R. Adams and J. A. Clarkson [2] and J. J. Gergen [8].

19 The proof of this theorem suggested itself by reason of certain similar ideas used by L. Tonelli in Serie Trigomometriche ( [17], Clh. 1X).




(3) Smn (,n Y) Km3(xn(x S)Kn(y, t)f (s, t)dsdt,

(4) Sm,n ("r" Y) C Gm (Xn S) Cln (y.7 t) f (s, t ) dsdt,

respectively. Also, using as before br(m,s) Km (xp,s) -GC.(x,s), we set

~ r s7r

(5 ) O9M (Xp Y) 41o ) m (Xp S) (n (y, t) f (s, t) dsdt,

mn (x, y) =

cnl (XPZ

S ) (n (yp t ) f (s, t ) dsd t,

(6) )?)(.r, y) = Of 3' O s(x),Cn(y, t)f(s,t)dsdt,

and show that the three integrals | ?(') (x, y) O mn (x, y) I and ?WI) (x, y) 0 as rn,m.-> co.

As for integral (3) 3 the convergence of ?() (x, y) to 0 follows from the uniform boundedness of I2 (X S) .20

To consider 0(?) (x, y) = ff ((x, S))4% (y, t)f (s, t)dsdt put

(8) Fn (s) ==FPn (S, y) =4' n (y, t) f (s, t) dt.

Then Fvn(s) is of bounded variation with respect to s in (0, 7r) for all y Tn (0,ir) and (n- 0,1,2, y ).

Proof. Let a be any subdivision of (0, 7r), o; 0 = sO < si < ... < Sk = 7r.

Then k k n

(9) E | n(Sv) -Fn (Sv-L) | - | (y, t) (f (sv, t) - f (sv-, t) )dt |

47r k CD | (yp t ) I Y I f (sv., t)-f (sv-:L. t) I dt.

O t=1

By (I ) ancd (2) in the definiition of b. v. T k

f f(sgv,t) -f(sv-1,t)l ? 2(t) for a.a. t and 72(t) U2(t)

(for all s) so that

k . 1

(10) P n (Sv)- Fn (Sv-1) | !4n (yp t) I TJ2 (t) dt. v=l O

20 By applying a well-known convergence theorem due to E. W. Hobson ([131,

vol. 2, pp. 422-4) we see that the conivergence of O,."' (x, y) to 0 is uniform. (See

7. 2).




Th-erefore k ~~~~~~~~~~/7r

(11) Vn(7) = Sllp | lyn(sv) -Fn(Sv-1), C?W CIb(y, t) I U2(t)dt.

eillce F. (s) 21 is of bounded variation and can be represented as Fn(s) = Fn (0) + Pn (s) -.YN (s), where the positive, monotonic non-decreasing functions, Pn (s) and A,, (s), are the positive and negative variations, respectively, of Fn (s) in (O, s).

Then

(12) ?~(x,(2) y) , Cf.(x, S) ((n(O) + Pn(s) - Nn (s) )ds

anid we can use the second mean value theorem getting

(13) ? (x2),(X y) == F (O) C7n((X S)dS + Pn(7r 0) Cfm(x, s)ds

Nit (7 -O) C. (x, s) ds, (O _ <7r, 0 _?q 7r)

([13], vol. l, p. 568).

Now it is well-known that 4 C. (x, s) ds j ? M, where M is independent

of m and x for all a, b such that 0 ? a < b ? 7r ([13], vol. 2, p. 510). Hence

(14) | 0(2) (X,y) I M( Fn(0)I +Pnr(-?0) +Nn(7r-0))

=M(I Fn(0)I + Vn(7r-O)).

Now I Fn(0)! Fn (SO)! + Vn (7) where go is such that f(so, t) is an integrable function of t (a. a. so). Since also VIn (,r -0) V,,(7r), then from inequalities (14) and (11)

(15) ?(2) (X, y) I M_( | P_ so) | -- (SO)+ ) ) inn

Vtr (, r

-( f n(y, t) f (so, t) dt + 2 ft n(y, t) I j2 (t) dt).

By Haar's theorem and the corollary to Lemma II the right side -> 0 as n -> oo. Hence I ? (2)(x,y) 1 - 0 as m, n r- o for all (x,y) in Q. Similarly for I 00) I and the theorem is proved.

inn

3. Remark on equiconvergence by rows and columns. A double 00 00

series Y A,,,, is said to be convergent by rows if all the series : Amn 2n=O, n=0 n=O

21 F (s) may be derined by (8) for oily a. a. s. For all exceptional s it may be defined as equal to either Fn(S + 0) or Fn(s- 0).




00 00

(m = 0, 1, 2, ) and the series E ( E A.m) converge. (Similarly for n=0 n10

convergence by columns). By a method similar to that used in the proof of Theoreml III the fol-

lowing equiconlvergence theorem can be proved.

THEOREM III'. If the integrable function f(x, y) is of b. v. T in Q, then its double SL series is equisummcable by r-ows aGnd columns with its double FCC series.

4. Sufficient conditions for convergence of the double SL series. Any one of the followilng conditions on the function f (x, y) is sufficient for the convergence of its double FCC series and also implies that the function is of b. v. T. [1] so that Theorem III is applicable.

(i) f(x,y) is of bounded variation (Hardy) ([10], [1])

(ii) f (x, y) is of bounded variation (Arzela) ([3], [7], p. 114),

(iii) f(x, y) satisfies any one of the many conditions sufficient for the convergence of the double Fourier series given by L. Tonelli ([17], pp. 450- 73). In all these conditions he demands that f(x, y) be of b. v. T. plus some other restrictions.

(iv) A condition recently introducedl by J. J. Gergen [8] is of the same character.

Hence

THEOREM IV. The double SL series of an integrable funtction f(x, y) is convergent at all points (x, y) in Q for which any one of conditions (i) to (iv) is satisfied. If (r, y) is a point of continuity of f (x,, y) the sumn of the series is f (x, y). If it is a point of ordinary discontinuity the sum is

4r(f(x + 0, y + 0)+ f(x + 0, y-O)+ f(x-0,y + 0) + f(x-0, y-O)).

PART IV. Modified Poisson Summability and its Application to the SL Series.

6. Modified Poisson summability method.

1. Let f(0) be integrable on (-7r, 7r). Applying the well-known Abel- Poisson summation method to Fourier series; we multiply its n-th term by rn, getting the series

2-b > (b7_it cos/7 nO 1 7

, sn nA rn f 0 A




which is absolultely and uniformly con+vergent for (- 7r 0 ? 7r) if (O ? r < 1). Then if f(0, r) h as a limit as r -> 1-0, the Fourier series is said to be summable by this mnethod. Fatou has proved that for any ilntegrable f(0) (2) lim f (0,; r) f (0) a. e. [6].

fl-41-6,

The analogous theorem is not valid for two dimensions. If it is assumed that f(0, p) log+ I f (0, 4) I is integrable, summability a. e. does follow [14]. It hlas been shown that the method works also for more general orthonormal systems {vn(0)}, (n 0, 1,2* ), if f(0) belongs to L2 and if suitable restrictions are imposed on the vn(9) ([15], pp. 186-94). (The SL system satisfies these restrictions.)

To get more general results the Poisson method may be modified as follows. Let v (0, r), (n 0, 1,2, ) be the function harmonic in the interior of the unit circle and reducing to v, (0) on the circumference, that is,

svr

(3) Vn (0 ,r) = fP(, r, s) v,n(s)ds, (n =O , 1 2,~ . . . 7r

where

(4) P(O,r,s)= 1 1_ -r2

2wpr 1 -2rcos (O-s) +r2 and

(5) lim vN(0, -) vn(0) a. e. (Fatou). r-*>1-0

(For v,, (s) = cos ns, ?n(0, r) = cos nO r1$ and similarly for the sine.) We theni form the series

o 00 7r

(6) E c;nv,,(O, r), cn - vn(s)f(s)ds (n ?'01,2, * ) n=o

(the modified Poissont (((m. P.)) series of f(O) formed with respect to {vt(0)}) and consider the limit as r -> 1-0. The orthogonal series

E c.v,n(O) of an integrcable function f(O) will be called summable by the n=0)

(in. P.) method if 00

(7) lim Y GnVn(0, r) r-*1-O n=0

exists. The method recluces to the Abel-Poisson for ordinary Fourier series. 00

Similarly we may define sumrnmability of the d'oub le seri'es E Cmn,Vm (6)Vn (Vn ) rn=0, 71=0

IIn this case we consider the limit

00

(8) lim E Cm,nvin (0 fr) vG (k, p), r-*1-O rn=o, n=O p-*l-e



ON DOU13LE STUIRM-LIOUVILLE SERIES. 631

where v,m (0, r) and v,. (p, p) are defined by (3). However it must be remarked that our method could be generalized in several ways to the case of two variables (see [4]). Carrying through the generalization for the double SL system by in-troducing double harmonic functions on a bicylinder we can prove equisummability (n. P.) of the double SL and FCC series for an integrable function withotut any further restriction (7. 2). (For the one variable case the proof is exceedinglv simple.) In 7. 3 we generalize some of our results to other orthonormal systems.

2. The modified Poisson method for orthogonal series of functions of L2 [5].

00 THEOREM 3. , cnvn,(0, r) converges absolutely and uniformly for all 0

n=o

in (- r, 7r) and r in any closed i.nterval in the interior of (0, 1) .22

Proof. By the Schwarz inequality 00 00 00

(9) E I C,,V,n(0, r) E Cn 2)( E Vn2 (0, r) . n0 =0 n=0

Then by (3) and Bessel's inequality applied to each factor on the right of (9) oo 7r > q

(10) E I COvn (0 r)j (f (s) ds) ( P2 (0 r, s) ds)i n3=0 r -T

27r 1 - r r)k (s) ds)-.

THEOREM 4. Let {v.n (6, r) }, (n - O, 1, 2, * ) be an orthonormal system complete uith respect to functions of class L2. Then the orthogonal series of f(0) with respect to this system is equisummable (m. P.) with its Fourier series.

The proof of this theorem, namely, that

(11) ~lim 1,Ec,,v, (0,r) - f(0, r)l 0, (-7 < 0-- x), rV-1-( n=0

00

is given in [5] by showing that the (m. P.) series v c0v.,, (6, r) is equicon- n=O

vergent with the Poisson integral f (0, r) for (O ? r < 1).

7. Application to SL series of functions of class L.

1. Equisummability of the simple SL and FC series. The following theorem is easily proved:

22 This proof of Theorem 3 seems to be more direct for real functions than Bergmnan's proof.




00

THEOREM V'. The (n. P.) series ( a,tOn(G, r) of any integrable func- n=o

tion f (0) is uniformly eqiiconvergent on (0, -r) with its Poisson integral (O ? r < 1). Uniform equisummability (in. P.) on (0, ir) of the SL series of f with its FC series follows.

Proof. Since the SL functions, {O,n (0) }, (n = 0, 1, 2, ), are defined on (0, r) we consider

^7r

(1) +pf(8, r) P(B, r, s)On(s)ds, 0

where

(2) P(O,r,s)=r{1-2rc7srs)+r2 s 1-2rcos(O+s)+r2V

If we suppose that f(0) is even, then

(3) f(O,r) f P(G,r,s)f(s)ds (cf. 6 (1))

and, in particular,

(4) cos nGrflJ P( r, s) cos nsds, 1= 4 P(Q r,s)ds. 0 0

Call n

(5)~~~~~~~Ot ?(0, r) anpv (0, r), 0

and

(6) u'(08,r) -bo + 0 bvcosvGrv.

We prove that

(7) lim I an(0Q r)-u'n(0O r) j0 for all 0 in (0, 7r) and (O ? r < 1). n-o00

Using (n(s, u) defined in 1 (10), we get by direct computation

7r 7r

(8) |jon (0,r) C'n (0n r) !I (Dn (SP u)P~ (0, r, s) f (u) duds I

7r 7r

= P(B, r, s) (J n (s, u)f (u) du) ds I (by Fubiini)

?f P(0,Pr,s)i f ,?(s,u)f(u)duIds

cr 1 -+r (DrnJn(s,u)f(u))dut ds. = 7 1-rJ J O

23 This corresponds exactly to what we have done for the cosine kernel in 1.




By Haar's theorem this last expression is less than e(1 + r)/(1 - r), for n > NV6, e an arbitrary positive number, which proves the theorem.

2. Equisumniahility of the double SL and FCC series. We niow prove the correspon(ling theorem for the double SL series of an integrable func- tion1 f(O, 4).

00 THEOREMr V. The (tn. 1P.) series I amntlon(0 r)'On (9 p) of any in-

mn,n=O

tegrable funictiont f(0, b) is equiconlvergent with its double Poisson1 integral over Q (O r K 1. 0 ? p < 1). Equtisummability (m. P.) over Q of the double SL series of f with its FCC ser ies follows.

Proof. Corresponding to the notation used ill (5) and (6), we let

mnn

(9) umn(t, g; r, p) Y. avpv (0, r)cp (0, p), V,,=o

and mu.

(10) a3,n (O., (; r, p) = Avib4e cos O'9 cos pu rvpf, V,,=o

where av1i, Avp and bvp are defined as in 1, and we prove that

(11) lim 1 cram.n(9,c ;rp) -P'mn(9,cO, p)! O= , Il?-+00, n l-o0

for all (0,O) in Q and (O? r < 1, 0 p < 1). Now

mn,n ro (12) 9.n n(9,p; r,p) = -a5 J P(, r, s)P(, p, t)v(s)q1(t) dsdt

fffJ KrJOJOJK.(s, u)K.(t, v)P(9, r, s)P(O, p, t)f (u, v)dudvdsdt

and similarly m,n

(13) u',mn(6,4;r,p) 2, Xvibv J P(, r, s)P(, p, t) cosvscos utdsdt

Jo f Cn (s, u)C. (t, v)P(9, r, s)P(+, p, t)f(u, v)dudvdsdt.

Let us set (in analogy to 3. 1)

7r r (14) AD(n,; r,p) f 4rn(s, u) Cn(t, v)P(9, r s) P(O p t)f (u, v) dudvdlsdt,

(15) A(2) (0, p) .,

( 15) "(,2 # + ,pn- Cm (s, u) (D. (t, v ) P (0, r, s) P ( , p, t ) f (u, v ) ditdvdsdt, (15) A~(G1,,O;r,,p)=f ... fms>nt)(r)(~tfu)iddd

(16) A(,"(6, m ; rn p) .. Uf. J. 6(s I )Zn(t, v)P(6, r, s)P(4, p, t)f(u, v)dudvdsdt.

8



634 JOSE1'HINE MITCHELLh

We wish to prove that the three integrals I All' A,-2)nI and I A(3)| 0 Mit i~~~~~~nn

as in,n-> oo for all (0,Qb) in Q and (O?r< 1,O?p < 1). To consider AM) (6, cp;r,p) proceed as in inequality (8) and then

(17) JAl^(QsbO;r,p)I

-JOW ff I P(O, r, s)j P(c, p, t)4 f fJ?m(s, u)>4n(t, v)f(u, v)dtdv j dsdt

c 1 1 + i 1 + Py X | X 3X b sU)'In (t v) f (ufv) dudv I dsdt. 7r wi-' 1-pJoJo Jo o

But we have proved in 5. 2 that the inner integral on the right side converges to 0 uniformly with respect to (s, t) in Q as mn, t -> co, so that

(18) jAl)(O.,sr,p)j < 1 +r 1+p E for m > Ne, mn ~ 1-r i-p n E an arbitrary positive number.

Considering A95M (0, c; r, p) we have that

19) IA (2 0,q P)I

7 7r ^ 7r 7r = I 54' f (u, v) (f Cim(s, u)P(O, r, s)ds) (f@n(t, v)P(+b, p, t)dt)dfudv

= ff f(u, v) | I f , O(s, u)P(0, r, s)ds) f ). (t, v)(+,p, t)dt I dudi.

Now by formula (4)

(20) C J m (s u)P(O,r,s)ds - = + cos vUcosvO rv I O ps~~~~~~~~~~~~~~~=1

<1 - 1-r

Secondly by Haar's theorem, for arbitrary positive E.

<7r

(21) 4'1 bn(t, v)P(b, p, t)dt I < E for it > N N =N( p)

so that

(22) 1 A( ) 0 ;r, ) !( r 1 -_ f (t, v)! dutdvl., for vn > N and all m.

Hence

(23) lim IA (2 (O,cp; r, p)!-0, for all (., p) and ( < 1,? 0 p < 1) in -*x, n --cx

Similarly for I AM (0,) ; r, p) and the theorem is proved.

3. Remark on the modified Poisson method for more general ortho-

normal systems. The applicability of this method is not limited to the SL



ON DOUBTLE STUR.M-LIOUVILLE SERIES. 635

orthoitorm-ial system. W e have seen in 6. 2 that the (m. P) series of any function- of class L2 with respect to a complete orthonormal system coincides w-ith the Poisson integral of f. If we assume that the function f(0) is of class L, we can prove the following theorem.

TI-TEOREM VI. Let {vqn(6) }, (n 0, 1,2 ), be a complete or-tho- n

normal system and let Ln(Q,s) = -vv(0)vv(s). If the condition v=()

h

(24) f Ln(0Q, s)ds I? M (, 7r h ?h r))24 -7z

(M independent of h, n and 0), is satisfied, then the (m. P) series of an integrable function f(0) is equiconvergent wvith its Poisson integral and the development of f (0) is equisummable (m. P.) with its Fourier series.

The proof of this theorem will be given on another occasion. This theorem can be generalized in various ways' to more variables.

BRYN MAWR COLLE(IE.

REFERENCES.

[1] C. R. Adams and J. A. Clarkson, " On definitions of bounded variation for functions of two variables," Transactions of the American Mathematical Society, vol. 35 (1933), pp. 824-854.

[2] - "Properties of functions f (.x, y) of bounded variation," Transactions of the American Mathematical Society, vol. 36 (1934), pp. 711-730.

[31 C. Arzela, " Sulle funzione di due variahili a variazione limitata," Rendiconti di Botogiba 1904-5 and 1907.

[4] S. Bergman, " A method for summation of series of orthogonial functions of two variables," Butlletin of the American Mathematical Society, vol. 47 (1941), p. 555, Abstract 47-7-302-t.

[5] , "A remark on the paper 'Sur les Functions analytiques de deux Variables complexes '," Journal of Mathematics and Physics (MIT), vol. 21 (1942), pp. 141-3.

24 S. Kaczmal z and H. Steinhaus ([15], p. 176), introducing the "Lebesgue

-7r f unctions," J Ln Ii(, s)J ds, (n~ = 0, 1,2,..) have proved that a sufficient con-

dition for the convergence a. e. of the orthogonal development of a function of class L2 is that these functions be bounded independently of n and 0, which is, of course, much stronger than (24). Similarly for the (C, 1) summability of an orthogonal develop-

7rr nient they assume for the arithmetic means, L*% (0,s), that I L*n(,s) I ds M

([15], p. 194).




[6] P. Fatou, " Series Trigonom6triques et Series de Taylor," Acta Mathematica, vol. 20 (1906), pp. 335-400.

[7] H. Geiringer, ' Trigonometrische Doppelreihen," Monatshefte fur Mathematik und Physik, vol. 29 (1918), pp. 65-144.

[8] J. J. Gergen, " Convergence criteria for double Fourier Series," Transactions of the American Mathematical Society, vol. 35 (1933), pp. 29-63.

[9] A. Haar. "Zur Theorie der orthogonalen Funktionensysteme," Mathematische Annalen, vol. 69 (1910), pp. 331-371.

[10] G. H. Hardy, " On double Fourier Series and especially those which repre- sent the double i-function with real and incommensurable parameters," Quarterly Journal of Mathematics, vol. 37 (1905-6), pp. 53-79.

[11] G. H. Hardy and J. E. Littlewood, "A maximal theorem with functioni- theoretic applications," Acda Mathematica, vol. 54 (1930), pp. 81-116.

[12] E. W. Hobson, "On a general convergence theorem and the theory of the representation of a function by series of normal functions," Proceedings of the London Mathematical Society, vol. ]l (1908-9), pp. 349-95.

[13] - , The Theory of Functions of a Real Variable, vol. 1 and 2 (1926). [14] B. Jessen, J. Marcinkiewicz and A. Zygmund, "Note on the differentiability

of multiple integrals," Funtdamenta Mathematicae, vol. 25 (1935), pp. 217-34. [15] S. Kaczmarz and H. Steinlaus, " Tlieorie der Orthogonalreihen," Monografje

Matematyczne, Tom VI, Warsaw (1935). [16] J. Marcinkiewicz and A. Zygmund, "On the summability of double Fouirier

Series," Funudamenta Mathematicae, vol. 32 (1939), pp. 122-132. [17] L. Tonelli, Serie Trigonometriche, Bologna (1928). [18] A. Zygmund, " Sur la theorie riemannienne de certains systemes orthogonaux,"

1, Studia Mathematica, vol. 2 (1930), pp. 97-170; II, Prace Mat. Fizyczne, vol. 39 (1932), pp. 73-117.

[19] , "On the differentiability of multiple integrals," Fundamenta Mathe- maticae, vol. 23 (1934), pp. 143-9.

[20] - , "Trigonometrical series," Monografje Matematyczne, Tom V, Warsaw (1935).



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