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Regularizations at the origin ofdistributions having prescribedasymptotic propertiesJasson Vindas aa Department of Mathematics , Ghent University , Krijgslaan 281Gebouw S22, 9000, Gent, BelgiumPublished online: 17 May 2011.

To cite this article: Jasson Vindas (2011) Regularizations at the origin of distributions havingprescribed asymptotic properties, Integral Transforms and Special Functions, 22:4-5, 375-382, DOI:10.1080/10652469.2010.541058

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Integral Transforms and Special FunctionsVol. 22, Nos. 4–5, April–May 2011, 375–382

Regularizations at the origin of distributions havingprescribed asymptotic properties

Jasson Vindas*

Department of Mathematics, Ghent University, Krijgslaan 281 Gebouw S22, 9000 Gent, Belgium

(Received 18 March 2010 )

By a regularization at the origin is meant an extension to Rn of a suitable distribution initially defined off the

origin. We study the regularizations of distributions when the generalized functions to be regularized haveprescribed asymptotic properties.A complete description of the asymptotic properties of the regularizationsis obtained.

Keywords: regularization; tempered distributions; quasiasymptotics; asymptotic behaviour of generalizedfunctions; slowly varying functions; asymptotically homogeneous functions

AMS Subject Classifications: 26A12, 41A60, 41A63, 44A35, 46F10

1. Introduction

Regularization of distributions refers to the problem of extending distributions that are a prioridefined on a smaller set. Typically, this situation arises when one constructs distributions out offunctions (or generalized functions) that have mild singularities at a point [9,10]. This is a veryimportant subject for both theoretical mathematics and mathematical physics.

In quantum field theory [3], regularization is also known as renormalization. The fundamentalproblem is often to find a suitable regularization in such a way that it is consistent with theexperimental considerations. Scaling asymptotic properties of distributions have been shown tohave a valuable role in this respect, as they bring new insights into the problem [1,23,24]. On theother hand, the relationship between regularizations and asymptotic properties of distributionsis also of importance from the point of view of pure mathematics, for instance, in areas such assingular integral equations [8], the study of boundary properties of holomorphic functions [5], orin the Tauberian theory for integral transforms [4,12,13,22,24]. In fact, as shown in recent studies[4,11,22], the asymptotic analysis of various integral transforms may be completely reduced to thestudy of asymptotic properties of regularizations of distributions; this is the case for the Laplaceand wavelet transforms.

In this article, we study the regularizations at the origin when the distribution to be regularizedpossesses prescribed quasiasymptotic properties [9,12,24] at either the origin itself or infinity.

*Email: jvindas@cage.Ugent.be

ISSN 1065-2469 print/ISSN 1476-8291 online© 2011 Taylor & FrancisDOI: 10.1080/10652469.2010.541058http://www.informaworld.com

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Our aim is to provide a full description of the asymptotic properties of the regularizations. Weemphasize that such a problem is essentially a Tauberian one and may be restated in terms ofMellin convolution-type integral transforms: quasiasymptotic behaviour is nothing but knowledgeof asymptotic information over (Mellin) convolution transforms for all kernels in a Schwartz spaceof test functions. Recently, this problem has been investigated in [5,18–20,22]; we shall giveextensions of those results, and in particular, we provide more detailed asymptotic informationfor critical degrees than that from [5]. We shall consider distributions with values in a Banachspace. The main results of this article are presented in Section 3.

2. Notation and preliminaries

The space E always denotes a fixed, but arbitrary, Banach space with norm ‖ · ‖. If h : R+ �→ E

and T : R+ �→ R+, we write h(λ) = o(T (λ)) if ‖h(λ)‖ = o(T (λ)), and similarly for the big O

landau symbol; let v ∈ E, we write h(λ) ∼ T (λ)v if h(λ) = T (λ)v + o(T (λ)). Let m ∈ Nn, we

use the notation ϕ(m) = (∂ |m|/∂xm)ϕ.

2.1. Spaces of distributions

The Schwartz spaces [14] of smooth compactly supported and rapidly decreasing test functionsare denoted by D(Rn) and S(Rn). We denote by D0(Rn) ⊂ D(Rn) and S0(Rn) ⊂ S(Rn) theclosed subspaces consisting of those functions vanishing at the origin together with all theirpartial derivatives of any order; we provide them with the relative topologies inhered from D(Rn)

and S(Rn), respectively.Let A(Rn) be a topological vector space of test function over R

n. We denote by A′(Rn, E) =Lb(A(Rn), E) the space of continuous linear mappings from A(Rn) to E with the topologyof uniform convergence over bounded subsets of A(Rn). We will mostly have A = D, S, D0,or S0. For vector-valued distributions, we refer to [15]. If f ∈ A(Rn) is a scalar generalizedfunction and v ∈ E, we denote by f v ∈ A′(Rn, E) the E-valued generalized function given by〈f (x)v, ϕ(x)〉 = 〈f (x), ϕ(x)〉v.

Observe that [6,9] (see also [7]) the elements of D0′(Rn, E) are precisely those distributions

defined on Rn \ {0} that admit extensions to R

n, while the elements of S0′(Rn, E) are those

distributions of slow growth at infinity defined on Rn \ {0} and having extensions to R

n as temperedE-valued distributions. So, if f0 ∈ S0′

(Rn, E) (resp.D0′(Rn, E)), there exists f ∈ S ′(Rn, E) (resp.

D′(Rn, E)) such that 〈f0, ϕ〉 = 〈f, ϕ〉, for each ϕ ∈ S0(Rn) (resp. D0(Rn)). Any f satisfying sucha property is called a regularization at the origin of f0 (in short, regularization), in accordance withthe classical terminology for regularizing divergent integrals [9,10]. Naturally, regularizations arenot unique and two of them may differ by a distribution of the form

∑|m|≤l δ

(m)vm, where δ is theDirac delta distribution.

2.2. Quasiasymptotics

The quasiasymptotics [9,12,24] measure the asymptotic behaviour of a distribution by asymptoticcomparison with Karamata regularly varying functions. Recall a measurable real-valued function,defined and positive on an interval (0, A] (resp. [A, ∞)), is called slowly varying [2,16] at theorigin (resp. at infinity) if

limλ→0+

L(aλ)

L(λ)= 1

(resp. lim

λ→∞L(aλ)

L(λ)= 1

), for each a > 0.

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In the next definition, A(Rn) is assumed to be a space of test functions on which the dilation is acontinuous operator; we are mainly concerned with A = D, S, D0, S0.

Definition 2.1 Let f ∈ A′(Rn, E) and let L be slowly varying at the origin (resp. at infinity).We say that

(i) f is quasiasymptotically bounded of degree α ∈ R at the origin (resp. at infinity) with respectto L in A′(Rn, E) if for each test function ϕ ∈ A(Rn)

supλ≤1

1

λαL(λ)‖〈f(λx), ϕ(x)〉‖ < ∞ (resp. sup

1≤λ

).

We write f(λx) = O(λαL(λ)) in A′(Rn, E) as λ → 0+ (resp. λ → ∞).(ii) f has quasiasymptotic behaviour of degree α ∈ R at the origin (resp. at infinity) with respect

to L in A′(Rn, E) if there exists g ∈ A′(Rn, E) such that for each test function ϕ ∈ A(Rn),

the following limit holds with respect to the norm of E

limλ→0+

1

λαL(λ)〈f(λx), ϕ(x)〉 = 〈g(x), ϕ(x)〉 ∈ E

(resp. lim

λ→∞

).

In such a case, we write

f(λx) = λαL(λ)g(x) + o(λαL(λ)) in A′(Rn, E) as λ → 0+ (resp. λ → ∞). (1)

We will also use the notation f(λx) ∼ λαL(λ)g(x) for (1).If A = S or D in (ii) of Definition 2.1, it is easy to show [9,12,24] that g must be homogeneous

with degree of homogeneity α, i.e. g(ax) = aαg(x), for all a ∈ R+. We refer to [5] for an excellentpresentation of the theory of multidimensional homogeneous distributions; such results are alsovalid for E-valued distributions.

Suppose that f0, g0 ∈ S0′(Rn, E) satisfy f0(λx) ∼ λαL(λ)g0(x) in S0′

(Rn, E). Then, g0 mustbe homogeneous of degree α over S0(Rn). Suppose now that α /∈ −n − N, applying [5, Theorem3.1 and Corollary 3.2], we conclude the existence of a unique regularization at the origin g ∈S ′(Rn, E), which is homogeneous of degree α. The case α = −n − p, p ∈ N, is slightly different;from [5, Corollary 3.3], we obtain a regularization g which is associate homogeneous of order 1and degree −n − p over S(Rn) (cf. [9, p. 74; 10; 17]); specifically, there are vm ∈ E, |m| = p,such that

g(ax) = a−n−pg(x) + a−n−p log a∑

|m|=p

δ(m)(x)vm, for each a > 0. (2)

Of course, the same considerations are true for D0′(Rn, E).

2.3. Asymptotically and associate asymptotically homogeneous functions

The functions to be introduced here will appear naturally in Section 3. The terminology in thescalar-valued case is from [18–21] (see also the de Haan theory in [2]); we remark that theseclasses of functions have already shown to be a tool of great importance in the study of asymptoticproperties of one-dimensional distributions. We list some of their basic properties in Section 4(Lemma 4.3).

Definition 2.2 Let c : (0, A) → E (resp. (A, ∞) → E), A > 0, be a continuous E-valuedfunction and let L be slowly varying at the origin (resp. at infinity). We say that

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(i) c is asymptotically homogeneous of degree γ ∈ R with respect to L if

c(aλ) = aγ c(λ) + o(L(λ)) as λ → 0+ (resp. λ → ∞), for each a > 0.

(ii) c is associate asymptotically homogeneous of degree 0 with respect to L if there exists v ∈ E

such that

c(aλ) = c(λ) + L(λ) log av + o(L(λ)) as λ → 0+ (resp. λ → ∞), for each a > 0.

(iii) c is asymptotically homogeneously bounded of degree γ ∈ R with respect to L if

c(aλ) = aγ c(λ) + O(L(λ)) as λ → 0+ (resp. λ → ∞), for each a > 0.

3. The main results: quasiasymptotic properties of regularizations

The following two theorems completely describe the asymptotic properties of arbitrary regular-izations when the distribution to be regularized has prescribed quasiasymptotic properties. Weonly state the tempered case, but the results are also valid if we replace everywhere below S byD (cf. Remark 3.3). In their proofs, we make use of three auxiliary lemmas, which are postponedfor Section 4.

Theorem 3.1 Let L be slowly varying at the origin (resp. at infinity) and let f0 ∈ S0′(Rn, E)

have the quasiasymptotic behaviour

f0(λx) = λαL(λ)g0(x) + o(λαL(λ)) in S0′(Rn, E) as λ → 0+ (resp. λ → ∞). (3)

Suppose that f ∈ S ′(Rn, E) is a regularization of f0. Then

(i) if α /∈ −n − N and g is the homogeneous regularization of g0, there exist d ∈ N and wm ∈ E,

|m| ≤ d, such that

f(λx) = λαL(λ)g(x) +∑|m|≤d

δ(m)(x)

λn+|m| wm + o(λαL(λ)) in S ′(Rn, E), (4)

(ii) if α = −n − p, p ∈ N, and g is a regularization of g0 satisfying (2), there exist d ∈ N, wm ∈E, |m| ≤ d, and associate asymptotically homogeneous E-valued functions cm, |m| = p,

satisfying

cm(aλ) = cm(λ) + L(λ) log avm + o(L(λ)), (5)

such that

f(λx) = L(λ)

λn+pg(x) +

∑|m|≤d,|m| =p

δ(m)(x)

λn+|m| wm +∑

|m|=p

δ(m)(x)

λn+pcm(λ) + o

(L(λ)

λn+p

)(6)

in the space S ′(Rn, E).

Proof The hypothesis (3) and Lemma 4.1 imply the existence of d ∈ N such that for any ρ ∈S(Rn) satisfying ρ(m)(0) = 0, |m| ≤ d ,

〈f(λx), ρ(x)〉 = λαL(λ)〈g(x), ρ(x)〉 + o(λαL(λ)).

Let η be a fixed test function such that η(x) = 1 on a neighbourhood of the origin and supp η ⊂B(0, 1), the ball of radius 1 centred at the origin. Given ϕ ∈ S(Rn), an arbitrary test function,

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we set

Tϕ(x) =∑|m|≤d

ϕ(m)(0)

m! xm,

its Taylor polynomial of order d . So, if we write ρ(x) := ϕ(x) − η(x)Tϕ(x), then we obtain

〈f(λx), ϕ(x)〉 = λαL(λ)〈g(x), ρ(x)〉 + 〈f(λx), η(x)Tϕ(x)〉 + o(λαL(λ))

= λαL(λ)〈g(x), ϕ(x)〉 +∑|m|≤d

ϕ(m)(0)

⟨f(λx) − λαL(λ)g(x),

xm

m! η(x)

⟩

+ o(λαL(λ)).

Thus, if set cm(λ) = (−1)|m|〈λ−αf(λx) − L(λ)g(x), xmη(x)/m!〉, we obtain the asymptoticformula

f(λx) = λαL(λ)g(x) +∑|m|≤d

δ(m)(x)λαcm(λ) + o(λαL(λ)), (7)

valid now in S ′(Rn, E). We study now the asymptotic properties of the E-valued functions cm. Foreach |m| ≤ d, we fix a test function ϕm such that ϕ(m)

m (0) = (−1)|m|, but ϕ(j)m (0) = 0 for j = m,

|j | ≤ d. At this point, we split the proof into cases (i) and (ii).Case (i): α /∈ −n − N.If we evaluate f(aλx) at ϕm(x), a > 0, use the homogeneity of g, the slowly varying property

of L, and apply (7), we obtain

aαλαL(λ)〈g(x), ϕm(x)〉 + aαλαcm(aλ) + o(λαL(λ))

= 〈f(aλx), ϕm(x)〉 =⟨f(λx),

1

anϕm

(x

a

)⟩

= λαL(λ)

⟨g(x),

1

anϕm

(x

a

)⟩+ a−n−|m|λαcm(λ) + o(λαL(λ))

= aαλαL(λ)〈g(x), ϕm(x)〉 + a−n−|m|λαcm(λ) + o(λαL(λ)).

Hence, for each |m| ≤ d , the E-valued function cm satisfies (i) in Definition 2.2 with γ = −α −n − |m|, and so, Lemma 4.3 gives the existence of wm ∈ E (some of them may be 0), for each|m| ≤ d, such that

cm(λ) = λ−α−n−|m|wm + o(L(λ)). (8)

Inserting (8) into (7), one gets (4). This completes the proof of the first case.Case (ii): α = −n − p.Observe that (2) shows that g is homogeneous when acting on test functions such that ϕ(j)(0) =

0 for |j | = p. Thus, if d < p, the previous argument shows that f satisfies indeed (6) with wm asbefore and cm identically 0. We suppose now that p ≤ d; if |m| = p, then the preceding argumentapplies also to show the existence of wm such that cm satisfy (8), which in turn implies

f(λx) = L(λ)

λn+pg(x) +

∑|m|≤d,|m| =p

δ(m)(x)

λn+|m| wm +∑

|m|=p

δ(m)(x)

λn+pcm(λ) + o

(L(λ)

λn+p

)(9)

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in S ′(Rn, E). We now analyse the behaviour of cm when |m| = p, it remains to establish (5).Evaluating (9) at ϕm (defined as before) and using (2), we have

L(λ)

(aλ)n+p〈g(x), ϕm(x)〉 + (aλ)−n−pcm(aλ) + o

(L(λ)

λn+p

)

= 〈f(aλx), ϕm(x)〉 =⟨f(λx),

1

anϕm

(x

a

)⟩

= L(λ)

λn+p〈g(ax), ϕm(x)〉 + (aλ)−n−pcm(λ) + o

(L(λ)

λn+p

)

= L(λ)

(aλ)n+p〈g(x), ϕm(x)〉 + L(λ) log a

(aλ)n+pvm + (aλ)−n−pcm(λ) + o

(L(λ)

λn+p

);

consequently, they satisfy the requirements. �

We now consider quasiasymptotic boundedness.

Theorem 3.2 Let L be slowly varying at the origin (resp. at infinity) and let f0 ∈ S0′(Rn, E)

be quasiasymptotically bounded of degree α at the origin (resp. at infinity) with respect to L inS0′

(Rn, E). Suppose that f ∈ S ′(Rn, E) is a regularization of f0. Then

(i) if α /∈ −n − N, there exist d ∈ N and wm ∈ E, |m| ≤ d, such that

f(λx) =∑|m|≤d

δ(m)(x)

λn+|m| wm + O(λαL(λ)) in S ′(Rn, E),

(ii) if α = −n − p, p ∈ N, there exist d ∈ N, wm ∈ E, |m| ≤ d, and asymptotically homo-geneously bounded E-valued functions cm, |m| = p, of degree 0 with respect to L

such that

f(λx) =∑

|m|≤d,|m| =p

δ(m)(x)

λn+|m| wm +∑

|m|=p

δ(m)(x)

λn+pcm(λ) + O

(L(λ)

λn+p

)in S ′(Rn, E).

Proof It is enough to set g = 0 and replace o by O in the arguments given in the proof ofTheorem 3.1. We leave the details of such modifications to the reader. �

Remark 3.3 Theorems 3.1 and 3.2 still hold if we replace S by D everywhere in the statements.Indeed, the proofs of these assertions are identically the same as the ones for the tempered case,but now making use of Lemma 4.2 instead of Lemma 4.1.

4. Auxiliary lemmas

We show in this section three lemmas that were used in Section 3.The following lemma is due to Drozhzhinov and Zavialov in the scalar-valued case [5, Lemma

2.1]; actually, a similar proof applies to the E-valued case. Denote by Sd(Rn), d ∈ N, the closed

subspace of S(Rn) consisting of functions such that all their derivatives up to order d vanish atthe origin; they are provided with the relative topology inhered from S(Rn).

Lemma 4.1 Let f ∈ S ′(Rn, E) and let L be slowly varying at the origin (resp. at infinity).

(i) Let g ∈ S ′(Rn, E). Suppose that the restrictions of f and g to S0(Rn) satisfy

f(λx) ∼ λαL(λ)g(x) in S0′(Rn, E).

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Integral Transforms and Special Functions 381

Then, there exists d ∈ N (large enough) such that the restriction of f to Sd(Rn) has the same

quasiasymptotic behaviour in the space S ′d(R

n, E).(ii) Suppose that the restriction of f to S0(Rn) is quasiasymptotically bounded at the origin (resp.

infinity) with respect to L in the space S0′(Rn, E), then there exists d ∈ N (large enough)

such that the restriction of f to Sd(Rn) is equally quasiasymptotically bounded with respect

to L (with the same degree) in the space S ′d(R

n, E).

Proof For each l ∈ N, define the norms

‖ρ‖l = max|m|≤lsupx∈Rn

(1

|x|2 + |x|2)l

|ρ(m)(x)|, (10)

for ρ ∈ S0(Rn). Obviously, these norms induce on S0(Rn) the same topology as the one inheredfrom S(Rn). For each l fixed, denote by Xl the completion of S0(Rn) with respect to the norm‖ · ‖l ; then, S0(Rn) = ⋂

Xl , the intersection having also topological meaning as a projectivelimit. The Banach–Steinhaus theorem implies that f has the same quasiasymptotic behaviour(resp. is equally quasiasymptotically bounded) over some Xl0 . But, clearly, if d is large enoughSd(R

n) ⊂ Xl0 . This shows the lemma. �

We have a similar assertion for the non-tempered case. Denote now Dd(Rn) = D(Rn) ∩

Sp(Rn), provided with the relative topology inhered from D(Rn).

Lemma 4.2 Let f, g ∈ D′(Rn, E) and let L be slowly varying at the origin (resp. at infinity).

(i) Suppose that the restrictions of f and g to D0(Rn) satisfy

f(λx) ∼ λαL(λ)g(x) in D0′(Rn, E).

Then, there exists d ∈ N (large enough) such that the restriction of f to Dd(Rn) has the same

quasiasymptotic behaviour in the space D′d(R

n, E).(ii) Suppose that the restriction of f to D0(Rn) is quasiasymptotically bounded at the origin (resp.

infinity) with respect to L in the space D0′(Rn, E), then there exists d ∈ N (large enough)

such that the restriction of f to Dd(Rn) is equally quasiasymptotically bounded with respect

to L (with the same degree) in the space D′d(R

n, E).

Proof Let D(B(0, 1)) ⊂ D(Rn), the subspace consisting of test functions supported by theclosed ball of radius 1 with centre at the origin. Denote Dd(B(0, 1)) = Dd(R

n) ∩ D(B(0, 1)),d ∈ N. Since any ϕ ∈ Dd(R

n) can be written as ϕ = ϕd + ϕ0, with ϕd ∈ Dd(B(0, 1)) and ϕ0 ∈D0(Rn), it is enough to show that the conclusions of the lemma are valid in one of the spacesD′

d(B(0, 1), E) for the restriction of f to Dd(B(0, 1)). For each l, let Yl be the completion ofD0(B(0, 1)) with respect to the norm (10); then, D0(B(0, 1)) = ⋂

Vl , as a projective limit. As inLemma 4.1, we conclude the existence of l0 such that f has the same quasiasymptotic behaviour(resp. is equally quasiasymptotically bounded) over Yl0 . Taking d large enough, Dd(B(0, 1)) ⊂Yl0 , which yields the result. �

One can obtain the precise asymptotic behaviour of asymptotically homogeneous and homo-geneously bounded functions of non-zero degree. The proof of the following lemma can be givenexactly as in the scalar-valued case [18] (see also comments in [20,21]); we choose to omit it andrefer the reader to the cited papers for a proof.

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Lemma 4.3 Let L be slowly varying at the origin (resp. at infinity).

(i) Assume that c is asymptotically homogeneous at the origin (resp. at infinity) of degree γ withrespect to L.(i.1) If γ > 0 (resp. γ < 0), then c(λ) = o(L(λ)).

(i.2) If γ < 0 (resp. γ > 0), then c(λ) = λγ w + o(L(λ)), for some w ∈ E.(ii) Assume that c is asymptotically homogeneously bounded at the origin (resp. at infinity) of

degree γ with respect to L.(ii.1) If γ > 0 (resp. γ < 0), then c(λ) = O(L(λ)).

(ii.2) If γ < 0 (resp. γ > 0), then c(λ) = λγ w + O(L(λ)), for some w ∈ E.

Acknowledgement

The author gratefully acknowledges the support given by a Postdoctoral Fellowship of the Research Foundation–Flanders(FWO, Belgium).

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