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Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


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Volume 20, Issue 5

Issues

A generalization of the Paley–Wiener theorem for Mellin transforms and metric characterization of function spaces

Carlo Bardaro / Paul L. Butzer / Ilaria Mantellini / Gerhard Schmeisser
Published Online: 2017-10-31 | DOI: https://doi.org/10.1515/fca-2017-0064

Abstract

We characterize the function space whose elements have a Mellin transform with exponential decay at infinity. This result can be seen as a generalization of the Paley–Wiener theorem for Mellin transforms. As a byproduct in a similar spirit, we also characterize spaces of functions whose distances from Mellin–Paley–Wiener spaces have a prescribed asymptotic behavior. This leads to Mellin–Sobolev type spaces of fractional order.

MSC 2010: Primary 44A05; 26A33; 46E35; 30G30; Secondary 30D20; 26D10

Key Words and Phrases: Mellin transforms; polar-analytic functions; Paley–Wiener spaces; Mellin–Hardy spaces; fractional Mellin derivatives; fractional Mellin–Sobolev spaces

Dedicated to Virginia Kiryakova on the occasion of her 65th birthday and to the 20th anniversary of FCAA – a top journal which Virginia founded and to which she devoted her whole energy to the benefit of all

1 Introduction

The classical Paley–Wiener theorem of Fourier analysis describes exactly the structure of the so-called Paley–Wiener spaces, whose elements are L2-functions with compactly supported Fourier transform, in terms of the Bernstein spaces comprising all functions fL2(ℝ) which have an analytic extension to the whole complex plane and are of exponential type (among the extensive literature, we quote here [33], [9], [30], [25], [31]). Analogues of the Paley–Wiener theorem were also obtained for other integral transforms (see e.g. [1], [2], [29]).

The present paper is concerned with further novel results in the broad realm of Mellin analysis and transform theory. As to the basic literature, see, e.g., [10], [11], [12], [13], [19], [23], [31]. The paper [10] was presented at the International Workshop “Transform Methods and Special Functions”, ably conducted by Virginia Kiryakova at Varna in August 23–30, 1996, in which fractional calculus also played an important role. Regarding the Paley–Wiener theorem for Mellin transforms, in [6] we have recently proven a version which is independent of Fourier analysis, by introducing the Mellin–Bernstein spaces, comprising all functions fXc2:={f:R+C:f()()c1/2L2(R+)}

which have an analytic extension to the Riemann surface of the (complex) logarithm and satisfy some suitable exponential type conditions. An equivalent and simpler formulation was given in [8], based on the notion of “polar-analytic” function, which avoids the use of the Riemann surfaces and analytic branches simply by considering functions defined on the half-plane ℍ := {(r, θ) : r > 0, θ ∈ ℝ}. This modified notion of analyticity leads naturally to the classical Cauchy–Riemann equations in polar form (see [16], [24]). We mention that another approach to a Paley–Wiener theorem for Mellin transforms was established by Z. Szmydt and her school in Cracow (see [18]). However, the approaches in [6], [8] are quite different from those in [26], [27] in which the authors start with functions (distributions) of compact support and characterize the resulting Mellin transform (related results are given in [21]); also see the review paper on Zofia Szmydt’s life [22]). The reason is that, other than in Fourier analysis, the inverse of a Mellin transform is not a Mellin transform. Therefore, when one characterizes compact support, it makes an essential difference if one does it for the original function or for its Mellin transform.

With the aim to characterize certain classes of functions in terms of the remainders of certain Gaussian type quadrature formulae, the authors in [17] proved an interesting generalization of the Paley–Wiener theorem, characterizing a space of functions for which the Fourier transform has an exponential decay at infinity. This has several interesting applications for quadrature formulae.

The main aim here is to extend our Paley–Wiener theorem for polar-analytic functions to the case of Mellin transforms with exponential decay at infinity. In order to do that, we introduce a function space, denoted by Hc(Ha), and prove that it constitutes the exact space of functions whose Mellin transform has exponential decay (see Section 3.2). We show that this space is “intermediate” with respect to two suitable Mellin–Hardy type spaces (for the general definition of a Mellin–Hardy space, see [8]).

As a first application of our result, we characterize in Section 4 certain function spaces in terms of a notion of “Mellin distance” which was introduced in [7] (see Section 2.3) and was used for obtaining precise estimates in certain approximate relations in Mellin analysis. In particular, we consider the space Hc(Ha) and a fractional Mellin–Sobolev type space.

In a forthcoming paper, we will apply our generalized Mellin–Paley–Wiener theorem for studying estimates of the remainders of quadrature formulae over the positive part of the real line.

2 Basic notions and preliminary results

In this section we present some basic definitions and preliminary results concerning Mellin transforms and related function spaces, in particular Mellin–Sobolev spaces, and a notion of distance.

2.1 Mellin transform and related concepts

Let C(ℝ+) be the space of all continuous, complex-valued functions defined on ℝ+, and let C(r)(ℝ+) be the space of all functions in C(ℝ+) with a derivative of order r in C(ℝ+). Analogously, by C(ℝ+) we mean the space of all infinitely differentiable functions. By Lloc1(ℝ+), we denote the space of all measurable functions which are integrable on every bounded interval in ℝ+.

For 1 ≤ p < +∞, let Lp(ℝ+) be the space of all Lebesgue measurable and p-integrable complex-valued functions defined on ℝ+ endowed with the usual norm ∥fp. Analogous notations hold for functions defined on ℝ.

For c ∈ ℝ, let us consider the space (see [11]) Xc={f:R+C:f()()c1L1(R+)}

endowed with the norm fXc:=f()()c11=0+|f(u)|uc1du.

More generally, let Xcp denote the space of all functions f: ℝ+→ ℂ such that f(⋅) (⋅)c−1/pLp (ℝ+) with 1 < p < ∞. In an equivalent form, Xcp is the space of all functions f such that (⋅)c f(⋅) ∈ Lμp(ℝ+), where Lμp(ℝ+) denotes the Lebesgue space with respect to the (invariant) measure μ (A) = ∫A dt/t for any measurable set A ⊂ ℝ+. Finally, for p = ∞, we define Xc as the space comprising all measurable functions f : ℝ+→ ℂ such that fXc:=supx>0xc|f(x)|<. For an extensive study of the case p = 2, see [13].

The Mellin translation operator τhc, for h ∈ ℝ+, c ∈ ℝ, f: ℝ+ → ℂ, is defined by (τhcf)(x):=hcf(hx)(xR+).

Setting τh:=τh0, we have (τhcf)(x)=hc(τhf)(x) and τhcfXc=fXc.

In the Mellin frame, the natural concept of a pointwise derivative of a function f is given by the limit of the difference quotient involving the Mellin translation; thus, if f′ exists, then limh1τhcf(x)f(x)h1=xf(x)+cf(x).

This gives the motivation for the following definition (see [11]): the pointwise Mellin differential operator Θc, or the pointwise Mellin derivative Θcf of a function f: ℝ+ → ℂ and c ∈ ℝ, is defined by Θcf(x):=xf(x)+cf(x)(xR+)

provided that f′ exists a.e. on ℝ+. The Mellin differential operator of order r ∈ ℕ is defined recursively by Θc1:=Θc,Θcr:=Θc(Θcr1).

For convenience, set Θr:=Θ0r for c=0 and Θc0:=I with I denoting the identity operator.

The Mellin transform of a function fXc is the linear and bounded operator defined by (see, e.g., [23], [19]) Mc[f](s)[f]Mc(s):=0+us1f(u)du(s=c+it,tR).

The inverse Mellin transform Mc1[g] of a function gL1(c + iℝ), is defined by Mc1[g](x):=xc2π+g(c+it)xitdt(xR+),

where in general Lp(c + i ℝ), for p ≥ 1, will mean the space of all functions g: c+i ℝ → ℂ with g(c +i⋅) ∈ Lp(ℝ).

The Mellin transform Mc2 of fXc2 is given by (see [13]) Mc2[f](s)[f]Mc2(s)=l.i.m.ρ+1/ρρf(u)us1du,

for s = c+it, in the sense that limρ+Mc2[f](c+it)1/ρρf(u)uc+it1duL2(c+iR)=0.

Analogously, we define the inverse Mellin transform of a function gL2(c + iℝ) by Mc2,1[g](x)=l.i.m.ρ+12π1/ρρg(c+it)xcitdt,

in the sense that limρ+Mc2,1[g](x)12πρρg(c+it)xcitdtXc2=0.

For any fXc2, there holds (see [13]) Mc2,1[Mc2[f]](x)=f(x)(a.e. on R+).

For functions in XcXc2, we have an important “consistency” property of the Mellin transform, namely Mc[f](c+it) = Xc2[f](c+it) for almost all t ∈ ℝ; see [13].

In what follows, we will need the concept of a Mellin–Paley–Wiener space and a Mellin inversion class. For c ∈ ℝ and σ > 0, the Mellin–Paley–Wiener space Bc,σp, p ∈ {1,2}, comprises all the functions in C(ℝ+) such that [f]Mcp (c+it) = 0 for almost all |t| > σ (for all |t| > σ if p = 1). The Mellin inversion class Mcp comprises all functions fXcpC(ℝ+) such that [f]McpL1(c + iℝ); see [7].

2.2 Mellin–Sobolev spaces

For any r ∈ ℕ, and p ≥ 1, we define the Mellin–Sobolev space Wcr,p as the space of all functions fXcp such that there exists gC(r−1)(ℝ+) with f = g a.e., g(r−1)ACloc(ℝ+) and ΘcrgXcp. Here ACloc(ℝ+) denotes the space of all locally absolutely continuous functions on ℝ+.

The space Wcr,p can be characterized as Wcr,p={fXcp:ΘcrfXcp}.

For r ∈ ℕ one has (see [11], Proposition 6) [Θcr]Mc(c+iv)=(iv)r[f]Mc(c+iv)(vR).

The same result also holds for p = 2, taking into account the general convolution theorem for Mellin transforms ([13, Lemma 3.1]). By the above result the Mellin–Sobolev spaces can be characterized as (p ∈ {1,2}) Wcr,p=fXcp:(iv)r[f]Mcp(c+iv)=[g]Mc(c+iv),gXcp.

Now we discuss the Mellin–Sobolev spaces of fractional order. Let α > 0 be fixed. In the following, for the complex numbers (−1)α and (iv)α, for v ∈ ℝ, we adopt the principal value of the logarithm, i.e., we take the principal argument in the interval [−π, π[. We put, for example, (1)α=exp(iαπ),(iv)α=|v|αexp(iα(π/2)sgn(v))=iαvα.

We define the “difference” of order α > 0 of f:ℝ+ → ℂ by means of the series Δhα,cf(x):=j=0αj(1)αjτhjcf(x)(x>0,h>0).

We remark that for any function fXc the fractional differences exist a.e. for any h > 0, and one has ([4]) |Δhα,cfXcfXcj=0αj.

Note that the series on the right-hand side of the previous inequality is convergent; see [15].

Definition 2.1

We define the (pointwise) Mellin fractional derivative of order α > 0 at a point x as the limit limh1Δhα,cf(x)(h1)α=:Θcαf(x)

provided it exists.

In [4] the strong fractional derivatives for functions fXc is introduced through the formula limh1Δhα,cf()(h1)αg()Xc=0,

and we set g : = s- Θcαf. Moreover, (see [4]) [s-Θcαf]Mc(c+iv)=(iv)α[f]Mc(c+iv)(vR).

Hence, if fXc has a strong fractional derivative of order α and also a pointwise fractional derivative ΘcαfXc, then [Θcαf]Mc(c+iv)=[s-Θcαf]Mc(c+iv)=(iv)α[f]Mc(c+iv).(2.1)

We define the Mellin–Sobolev space Wcα,p(ℝ+), p ∈ {1,2}, as Wcα,p(R+):={fXcp:Θcαfexists a.e. and ΘcαfXcp}.

We have the following proposition:

Proposition 2.1

Let fMcp, p ∈ {1,2}, and let α > 0 be fixed. Denoting, for brevity, by φ the Mellin transform of f, if vα φ (v) ∈ L1(ℝ), then Θcαf exists and Θcαf(x)=(i)α2π+vαφ(v)xcivdv,x>0.

Proof

By the inversion theorem for Mellin transforms (see [11]) we have, for h > 0, Δhα,cf(x)(h1)α=xc2π+φ(v)1(h1)αj=0αj(1)αjhvjixivdv=12π+φ(v)xciv(hiv1)α(h1)αdv.

Now, taking into account that for any complex number z one has |zα| = |z|α, it is not difficult to verify that (hiv1h1)α2αsin((vlogh)/2)h1α2α|v|α.

Thus, passing to the limit h→ 1 and taking into account that limh1(hiv1)α(h1)α=(iv)α,

we obtain, by the Lebesgue theorem of dominated convergence, that (Θcαf)(x)=(i)α2π+vαφ(v)xcivdv,

and so the assertion follows. □

For the role of Mellin transform theory in the broad area of fractional calculus together with a variety of important applications in the realm of special functions, the reader may consult [20].

2.3 A notion of distance

In [7] we introduced a notion of distance associated with the Mellin transform. Here we recall our approach and some basic facts. For q ∈ [0, +∞], let Gcq be the linear space of all functions f:ℝ+ → ℂ that have the representation f(x)=12π+φ(v)xcivdv(x>0),

where φL1(ℝ) ∩ Lq(ℝ).

The space Gcq is endowed with the norm [|f|]q:=φLq(R)=+|φ(v)|qdv1/q.

The above norm induces the metric distq(f,g):=[|fg|]q,f,gGcq.

In the case q = 2, we have dist2(f,g)=2πfgXc2,

i.e., our distance reduces to the Euclidean distance in Xc2, up to the factor 2π.

As a consequence of the Mellin inversion formula, functions f for which [f]Mc(c+i⋅) ∈ L1(ℝ) ∩ Lq(ℝ) belong to Gcq. In particular if fMc1, then fGc and [[f|]=supvR|[f]Mc(c+iv)|.

The following representation theorem holds (see [7, Theorem 1]).

Theorem 2.1

For any fGcq, we have distq(f,Bc,σp)=|v|σ|φ(v)|qdv1/q(1q<),

and if φ is continuous, then dist(f,Bc,σp)=sup|v|σ|φ(v)|.

As a corollary, we obtain a representation of the distance of Mellin derivatives (see [7, Corollary 1]):

Let fGcq with vkφL1(ℝ) ∩ Lq(ℝ). Then for every p ∈ {1,2}, we have distq(Θcαf,Bc,σp)=|v|σ|vαφ(v)|qdv1/q(1q<),

and if φ is continuous, then dist(Θcαf,Bc,σp)=sup|v|σ|vαφ(v)|.

Analogously, using Proposition 2.1, we have:

Let fGcq with vαφL1(ℝ) ∩ Lq(ℝ). Then for every p ∈ {1,2}, we have distq(Θcαf,Bc,σp)=|v|σ|vαφ(v)|qdv1/q(1q<),

and if φ is continuous, then dist(Θcαf,Bc,σp)=sup|v|σ|vαφ(v)|.

3 Polar-analytic functions and Mellin–Paley–Wiener type theorems

3.1 Basic definitions and results

Let ℍ := {(r,θ) ∈ ℝ+ × ℝ} be the right half-plane and let 𝓓 be a domain in ℍ. We begin with the following definition (see [8]):

Definition 3.1

We say that f: 𝓓 → ℂ is polar-analytic on 𝓓 if for any (r0, θ0) ∈ 𝓓 the limit lim(r,θ)(r0,θ0)f(r,θ)f(r0,θ0)reiθr0eiθ0=:(Dpolf)(r0,θ0)

exists and is the same howsoever (r, θ) approaches (r0, θ0) within 𝓓.

For a polar-analytic function f, we define the polar Mellin derivative as Θcf(r,θ):=reiθ(Dpolf)(r,θ)+cf(r,θ).

Remark 3.1

This modified notion of analyticity, arising by treating polar coordinates as cartesian coordinates, leads naturally to the classical Cauchy-Riemann equations when written in their polar form, i.e., f = u + iv with u,v: 𝓓 → ℝ is polar-analytic on 𝓓 if and only if u and v have continuous partial derivatives on 𝓓 that satisfy the differential equations uθ=rvr,vθ=rur.(3.1)

For the derivative Dpol, we easily find that (Dpolf)(r,θ)=eiθ[ru(r,θ)+irv(r,θ)]=eiθr[θv(r,θ)iθu(r,θ)].

Also note that Dpol is the ordinary differentiation on ℝ+. More precisely, if φ (⋅) : = f(⋅, 0) then (Dpolf)(r,0) = φ′(r).

Moreover, for θ = 0 we obtain the known formula for φ: Θcφ(r)=rφ(r)+cφ(r).

When g is an entire function, then f: (r, θ) ↦ g(re) defines a function f on ℍ that is polar-analytic and 2π-periodic with respect to θ. The converse is also true. However, there exist polar-analytic functions on ℍ that are not 2π-periodic with respect to θ. A simple example is the function L(r, θ) := log r + , which is easily seen to satisfy the differential equations (3.1).

Definition 3.2

For c ∈ ℝ, T > 0 and p ∈ [1, +∞[ the Mellin–Bernstein space Bc,Tp comprises all functions f: ℍ → ℂ with the following properties:

  1. f is polar-analytic on ℍ;

  2. f(⋅, 0) ∈ Xcp;

  3. there exists a positive constant Cf such that |f(r,θ)|CfrceT|θ|((r,θ)H).

For the reader’s guidance, we mention that in the notations Bc,Tp, introduced already in [13, § 5.1] for p = 2, and Bc,Tp, the letter B shall remind on bandlimited and the letter ℬ on Bernstein.

As proved in [8], if fBc,Tp, then we have:

  • (j)

    f(⋅, θ) ∈ Xcp for all θ ∈ ℝ and f(,θ)XcpeT|θ|f(,0)Xcp;

  • (jj)

    limr0rcf(r,θ)=limr+rcf(r,θ)=0 uniformly with respect to θ on all compact subinterval of ℝ.

The Paley–Wiener theorem for the space Bc,T2 is the following statement (see [8, Theorem 4]):

Theorem 3.1

(Paley-Wiener). A function φXc2 belongs to the Mellin–Paley–Wiener space Bc,T2 if and only if there exists a function fBc,T2 such that f(⋅, 0) = φ(⋅).

A proof of the hard part of Theorem 3.1 will also follow from Theorem 3.2 below. Our approach to the generalized Paley–Wiener theorem, which involves polar-analytic functions, is very useful in order to define in a simple way a Hardy space in the Mellin frame. For c ∈ ℝ and p ∈ [1,+∞[, we recall that the norm in Xcp is defined by φXcp=(0+|φ(r)|prcp1dr)1/p.

For a > 0, we define a strip ℍa as Ha:=(r,θ)R+×]a,a[.

Definition 3.3

Let a, c, p ∈ ℝ with a > 0 and p ≥ 1. The Mellin–Hardy space Hcp(ℍa) comprises all functions f : ℍa → ℂ with the following properties:

  1. f is polar-analytic on ℍa;

  2. f(⋅, θ) ∈ Xcp for each θ ∈ ] − a, a [;

  3. there holds fHcp(Ha):=sup0<θ<a(f(,θ)Xcpp+f(,θ)Xcpp2)1/p<+.

When a ∈ ]0, π], we can associate with each function fHcp(ℍa) a function g analytic on the sector 𝓢a := {z ∈ ℂ : |arg z| < a} by defining g(re) := f(r, θ). The collection of all such functions constitutes a Hardy type space Hp(𝓢a), which may be identified with Hcp(ℍa).

In [14] the authors considered a Hardy space Hp(Sa) of functions analytic on the strip Sa := {z ∈ ℂ : |ℑz| < a}. The Hardy space Hcp(ℍa) has been designed in such way that if gHp(Sa) and f(r,θ):=rceicθg(logr+iθ),

then fHcp(ℍa) and conversely, if fHcp(ℍa) and g(x+iy):=ec(x+iy)f(ex,y),

then gHp(Sa).

3.2 A generalization of the Paley-Wiener theorem for Mellin transforms

In this section we aim at the generalization of the Paley–Wiener theorem for the Mellin transform which will later be applied for characterizing function spaces in terms of the distance from a Mellin–Bernstein space. The Fourier counterpart was proved in [17]. We need the following space Hc(ℍa), which lies between two Mellin-Hardy spaces.

Definition 3.4

Let a, c ∈ ℝ with a > 0 be fixed numbers. The class Hc(ℍa) comprises all functions f : ℍa → ℂ with the following properties:

  1. f is polar-analytic on ℍa;

  2. limr→0+ rcf(r, 0) = limr→+∞ rcf(r, 0) = 0;

  3. for every ε ∈ ]0, a[ there exists a constant K(f, ε) such that |f(r,θ)|rcK(f,ε)

    for all (r, θ) ∈ ℍaε;

  4. for every θ ∈ ]−a, a[ and all t ∈ ℝ, Ic(f,θ,t):=limR+1/RRf(r,θ)rc+it1dr

    exists and |Ic(f, θ, t)| ≤ K(f) with a constant depending on f only.

For the subsequent considerations, we need some lemmas concerning the (C, 1)-summability of the L2-Mellin inversion and a result on the inversion of the Cauchy principal value of a Mellin transform.

Lemma 3.1

Let G(c + i ⋅) be a function in L2(c + i ℝ). Then, denoting by gXc2 the inverse Mellin transform of G, one has g(r)=limρ+12πρρ(1|t|ρ)G(c+it)rcitdt(a.e. rR+).

Proof

First of all, by [13, Theorem 2.9], the inverse Mellin transform gXc2 exists. Moreover, if we employ the Mellin–Fejer kernel, defined by (see [11]) Fρc(r):=12πρrc(rρi/2rρi/2logr)2,r1andFρc(1)=ρ2π,

the Mellin convolution (Fρcg)(x):=0Fρc(xu)g(u)duu

converges almost everywhere to g. Furthermore, using a Parseval type equation for Mellin convolutions (see [11, Theorem 9]) and the fact that the Mellin transform of Fρc is given by [Fρc]Mc(c+it)=(1|t|ρ),0|t|ρ

and [Fρc]Mc(c+it)=0,|t|>ρ,

one has g(r)=limρ+12πρρ(1|t|ρ)G(c+it)rcitdt( a.e. rR+),

which is the assertion.□

The second lemma is as follows:

Lemma 3.2

Suppose that ϕ : ℝ+ → ℂ is integrable on every compact subinterval and G(c+it):=limR+1/RRϕ(r)rc1+itdr

exists for each t ∈ ℝ and is integrable on compact subintervals. Then, ϕ(r)=limρ+12πρρ(1|t|ρ)G(c+it)rcitdt

almost everywhere for r ∈ ℝ+.

In the Fourier case this lemma is stated and proved in [28, Theorem 120] and in [33, Theorem 10.3]. Both proofs are quite long with several intermediate results. For this reason, we made no attempts to design an intrinsic Mellin type proof, but leave the deduction of Lemma 3.2 to the reader, using the classical substitution from the Fourier to the Mellin instance. (Note: The modern literature on integral transforms prefers an Lp-frame. However, as soon as contour integration of analytic functions is involved, Cauchy principal values occur in a natural way. Therefore it would be desirable to find a new and short approach to Lemma 3.2 (without substitutions) or at least to its Fourier or Laplace version.)

Combining Lemma 3.1 and Lemma 3.2, we obtain the following basic lemma whose Fourier version was given in [17, Lemma 2.1].

Lemma 3.3

Let ϕ : ℝ+ → ℂ be integrable on every compact subinterval of+, and suppose that Mc[ϕ](c+it):=limR+1/RRϕ(r)rc+it1dr

exists for all t ∈ ℝ. If Mc [ϕ] ∈ L2(c + i ℝ), then ϕXc2 and Mc[ϕ](c + it) = Mc2[ϕ](c + it) almost everywhere for t ∈ ℝ.

Proof

If Mc[ϕ] ∈ L2(c + i ℝ), then, by a Plancherel type theorem for the Mellin transform (see [13, Theorem 2.9]), there exists a function ψXc2 such that Mc2[ψ] = Mc[ϕ]. Employing Lemma 3.1, we conclude that ψ(r)=limρ+12πρρ(1|t|ρ)Mc[ϕ](c+it)rcitdt(a.e. rR+).

On the other hand, by Lemma 3.2, we have ϕ(r)=limρ+12πρρ(1|t|ρ)Mc[ϕ](c+it)rcitdt(a.e. rR+).

Hence ϕ(r) = ψ(r) almost everywhere for r ∈ ℝ+. Now the assertion follows immediately.□

Remark 3.2

Using the notation of Lemma 3.3, we may express the quantity figuring in statement (d) of Definition 3.4 as Ic(f,θ,t)=Mc[f(,θ)](c+it).

With the help of Lemma 3.3, we can easily see that Hc(ℍa) constitutes a normed linear space.

Proposition 3.1

Let fHc(ℍa). By introducing fHc(Ha):=sup{|Ic(f,θ,t)|:θ]a,a[,tR},

the class Hc(ℍa) becomes a normed linear space.

Proof

As the only non-trivial assertion, we have to show that if fHc(ℍa) and fHc(Ha) = 0, then f(r, θ) = 0 for all (r, θ) ∈ ℍa. Now, if fHc(Ha) = 0, then, clearly, Ic(f, θ, t) = 0 for all t ∈ ℝ. In view of Lemma 3.3, we obtain that Mc2 [f(⋅, θ)](c + it) = 0 almost everywhere for t ∈ ℝ, and application of the the Mellin–Plancherel theorem (see [13, Lemma 2.6]) shows that f(r, θ) = 0 almost everywhere for r ∈ ℝ+. Since a polar-analytic function is continuous, we conclude that f(r, θ) = 0 for all (r, θ) ∈ ℍa.□

Next we want to show that in Definition 3.4, statement (b) can be generalized by employing (a) and (c).

Proposition 3.2

Let f : ℍa → ℂ satisfy conditions (a)-(c) of Definition 3.4. Then, limr0+rcf(r,θ)=limr+rcf(r,θ)=0

uniformly with respect to θ on all compact subintervals of ] − a, a [.

Proof

For x ∈ ℝ and y ∈ ]−a, a[, we define F(x+iy):=ec(x+iy)f(ex,y).

Then (a) of Definition 3.4 implies that F is analytic in the strip Sa := {z ∈ ℂ : |ℑ z|< a} and (c) shows that for every ε ∈]0, a [ it is bounded on Saε. Furthermore, by (b) we have F(x) → 0 as x → ± ∞. Then, by a theorem of Montel (see [9, Theorem 1.4.9]), it follows that F(x + iy) → 0 as x → ± ∞ uniformly with respect to y on compact subintervals of ]−a, a[. Expressing this conclusion in terms of f, we arrive at the desired result.□

Now we want to compare Hc(ℍa) with Hc1(ℍa).

Proposition 3.3

We have Hc1(Ha)Hc(Ha) and fHc(Ha)2fHc1(Ha).(3.2)

Proof

Let fHc1(ℍa). First we have to show that f satisfies conditions (a)–(d) of Definition 3.4. Clearly, (a) is guaranteed by the definition of Hc1(ℍa). Furthermore, (b) and (c) follow from Propositon 2 and Proposition 4 in [8], respectively. As regards (d), we note that Ic(f, θ, t) exists by the hypotheses of Hc1(ℍa) and |Ic(f,θ,t)|limR+1/RR|f(r,θ)|rc1dr=f(,θ)Xc12fHc1(Ha).

This not only verifies (d) but proves (3.2) as well.

The inclusion in (3.2) is strict since it can be shown that f(r,θ):=rceicθa+i(logr+iθ)

belongs to Hc(ℍa) but fHc1(ℍa).□

We can also compare Hc(ℍa) with a corresponding Hardy type space for L2-norms. The following proposition implies that Hc(ℍa) is a subspace of Hc2(ℍaε) for every ε ∈ ]0, a[.

Proposition 3.4

Let fHc(ℍa). Then, for all α ∈ ]−a, a[, we have |Mc[f(,α)](c+it)|e|t|(a|α|)fHc(Ha),(3.3)

f(⋅, α) ∈ Xc2 and f(,α)Xc212π(a|α|)fHc(Ha).(3.4)

Proof

We proceed in a way similar to the proof of [8, Theorem 5]. For fHc(ℍa) and t ∈ ℝ, we consider the function g defined by g(r,θ):=e(c1+it)(logr+iθ)f(r,θ),

which is polar-analytic on ℍa. For R > 1 and α ∈ ]−a, a[\ {0}, let γ be the positively oriented rectangular curve with vertices at (1/R, 0), (R, 0), (1/R, α) and (R, α). By an integral theorem for polar-analytic functions (see [8, Proposition 1]), we have γg(r,θ)eiθ(dr+irdθ)=0.

This equation may be written in terms of ordinary integrals as 1/RRg(r,0)dr=1/RRg(r,α)eiαdr+I(1/R,t)I(R,t),(3.5)

where, for any r > 0, I(r,t)=0αg(r,θ)ireiθdθ=irc+it0αe(tic)θf(r,θ)dθ.

Using statement (d) of Definition 3.4, we find that the first two integrals in (3.5) converge as R → +∞ while the last two expressions approach 0 as a consequence of Proposition 3.2. More precisely, in the notation of Definition 3.4, the limit R → +∞ in (3.5) leads to Ic(f,0,t)=eiα(c+it)Ic(f,α,t)

and so |Ic(f,0,t)|=eαt|Ic(f,α,t)|.(3.6)

Given t ≠ 0, we may choose in (3.6) the sign of α such that α t > 0. Noting that |Ic(f, α, t)| ≤ fHc(Ha) and letting |α| → a, we obtain |Ic(f,0,t)|ea|t|fHc(Ha).

Combining this with (3.6), we arrive at |Ic(f,α,t)|e|t|(a|α|)fHc(Ha).(3.7)

Since Ic(f, α, t) = Mc[f(⋅, α)](c + it), we have proved (3.3).

Using Lebesgue’s theorem of dominated convergence, we conclude from (3.7) that Ic(f, α, ⋅) ∈ L2(ℝ) and Ic(f,α,)L2(R)1a|α|fHc(Ha).

Now Lemma 3.3 implies that f(⋅, α) ∈ Xc2 for |α| < a. Finally, the same lemma and a Parseval type theorem for Mellin transform (see [13, Lemma 2.6]) lead us to inequality (3.4).□

Now we are ready for the announced generalization of the Paley–Wiener theorem for the Mellin transform.

Theorem 3.2

A continuous function ϕ : ℝ+ → ℂ is the restriction to+ of a function fHc(ℍa) if and only if Mc[ϕ] exists and |Mc[ϕ](c+it)|Cea|t|(tR),(3.8)

with a constant C that may be taken to be fHc(Ha).

Proof

If ϕ is the restriction to ℝ+ of a function fHc(ℍa), then Proposition 3.4 applies and (3.3) for α = 0 shows that (3.8) holds with C = fHc(Ha).

Conversely, suppose that Mc[ϕ] exists and (3.8) holds with C = fHc(Ha). Then, as a function of t, the expression Mc[ϕ](c + it)eiαt belongs to L1(ℝ)∩ L2(ℝ) whenever |α| < a. Therefore, f(r,θ):=12π+Mc[ϕ](c+it)(reiθ)citdt

exists for (r, θ) ∈ ℍa. By Lemma 3.3 and the Mellin inversion theorem in Xc2 (see [13, Theorem 2.9]), we obtain that f(r, 0) = ϕ (r) almost everywhere on ℝ+. Since Mc[ϕ](c + i ⋅) ∈ L1(ℝ), we conclude that f(⋅, 0) is continuous. Hence f(r, 0) = ϕ(r) for every r ∈ ℝ+.

Now we want to show that f satisfies (a)–(d) of Definition 3.4. By a refinement of the proof of [13, Theorem 4], one can verify that f is polar-analytic in ℍa. For the reader’s convenience we give details. Let (r0, θ0) be any point in ℍa. Denote by Q a sufficiently small closed rectangular domain centered at (r0, θ0) such that Q ⊂ ℍa. We consider the difference quotient f(r,θ)f(r0,θ0)reiθr0eiθ0=12π+Mc[ϕ](c+it)(reiθ)cit(r0eiθ0)citreiθr0eiθ0dt.(3.9)

For (r, θ) ∈ Q, the limit (r, θ) → (r0, θ0) carried out inside the integral leads to an ordinary differentiation of zcit with respect to z at the point z0 := r0 e0 and the integral with the derivative in place of the difference quotient exists. However, we have to justify that the limit and the integration can be interchanged. For this purpose, we first note that the difference quotient on the right-hand side of (3.9) can be expressed as (reiθ)cit(r0eiθ0)citreiθr0eiθ0=(c+it)01(r0eiθ0+s(reiθr0eiθ0))c1itds,

which implies that (reiθ)cit(r0eiθ0)citreiθr0eiθ0c2+t2sup(r,θ)Q(reiθ)c1+itc2+t2r1c+1eθ1|t|

for some r1 > 0 and θ1 ∈ ]0, a[. Thus, for (r, θ) ∈ Q the absolute value of the integrand in (3.9) is bounded by Cc2+t2r1c+1e(aθ1)|t|,

which is integrable with respect to t over ℝ. Hence the desired interchange is guaranteed by Lebesgue’s theorem of dominated convergence. This completes the proof of property (a).

Next we note that (b) is a consequence of the Riemann–Lebesgue lemma. Furthermore, using (3.8), we see that |f(r,θ)|12π+|Mc[ϕ](c+it)|rceθtdtCrc2π+e|t|(a|θ|)dt=rcCπ(a|θ|),

which shows that (c) also holds.

It remains to verify (d). By the hypothesis Ic(f, 0, t) = Mc[ϕ](c + it) exists. Now performing a contour integration as in the proof of Proposition 3.4 and noting that Proposition 3.2 is applicable to f, we conclude that Ic(f, α, t) also exists for |α| < a and (3.5) holds. Combined with (3.8), this yields |Ic(f,α,t)|Ce|t|(a|α|),

which shows that (d) holds with K(f) = C.□

Remark 3.3

Note that the “hard part” of Theorem 3.1 (i.e., a function fBc,T2 is such that f(⋅, 0) ∈ Bc,T2) can now be deduced from Theorem 3.2. Indeed, let fBc,T2. First assume in addition that f(⋅, 0) ∈ Xc. Then Mc[f(,0)]=Mc[f(,0)]andfHc(Ha) for every a > 0. By [8, Theorem 3], we conclude that fHc(Ha)sup|θ|<a0+|f(r,θ)|rc1drsup|θ|<af(,θ)XceaTf(,0)Xc.

Now by Theorem 3.2 we have |Mc[f(,0)](c+it)|ea|t|eaTf(,0)Xc

for every a > 0. Letting a → + ∞, we find that Mc[f(,0)](c+it)|=0whenever|t|>T,

that is f(⋅, 0) is Mellin bandlimited to [−T, T].

We still have to get rid of the additional assumption that f(⋅, 0) ∈ Xc.

Again, let fBc,T2. Since the space Xc2Xc is dense in Xc2, we can find a sequence of functions FnBc,T2, where n ∈ ℕ, such that fn(⋅) := Fn(⋅, 0) ∈ Xc2Xc and fn()f(,0)Xc20(n+).

Using the Mellin–Plancherel theorem (see [13]), we also have [fn]Mc2(c+i)[f]Mc2(c+i)L2(c+iR)0(n+).

From the first part of the proof, applying to functions in Xc, we know that [fn]Mc2(c+i)[f]Mc2(c+i)L2(c+iR)2=([fn]Mc2(c+i)[f]Mc2(c+i))χ[T,T]L2(c+iR)2+[f]Mc2(c+i)χ{|t|>T}L2(c+iR)2

for every n ∈ ℕ, from which we deduce that [f]Mc2(c+i)χ{|t|>T}L2(c+iR)2=0,

and so [f]Mc2(c+it)=0 almost everywhere for |t| > T.

In order to construct a suitable sequence with the above properties, for a given function fBc,T2, we can approximate f(⋅, 0) by a sequence of Mellin bandlimited functions in Xc2 with a prescribed asymptotic behaviour for r → 0 and r → + ∞. To be more precise, we construct a sequence of functions FnBc,T2 such that rc|Fn(r,0)|=O((logr)2)(r0,r+)

and Fn(,0)f(,0)Xc20(n+).

We can define, for example, Fn(r,θ):=f(anreiθ)sinc3(bnlog(reiθ)),

where (an) and (bn) are two numerical sequences with an → 1 and bn → 0+; for the details, see [5] and also [8] for a similar construction. The prescribed asymptotic behaviour of Fn ensures that Fn(⋅, 0) ∈ Xc2Xc.□

4 Characterizations of distances by function spaces

We are going to establish a characterization of the distances from Mellin–Paley–Wiener spaces by functions spaces guaranteeing a certain regularity.

For doing this, we have to characterize prescribed distances by spaces. As a general assumption, we may take fMc1.

We begin with the following characterization which is an immediate consequence of Theorem 2.1.

Theorem 4.1

(exactness). Let fMc1. Then fBc,σ1dist(f,Bc,τ1)=0,

for all τ > σ.

The following theorem characterizes the exponential rate of convergence of the distance from the space Bc,σ1. Here, as before, for a function f : ℝ → ℂ, we set φ(⋅) = f(⋅, 0).

Theorem 4.2

(exponential rate). Let φMc1. Then φ is the restriction to+ of a function fHc(ℝa) if and only if dist(φ, Bc,σ1) = 𝓞(e) as σ → + ∞.

Proof

This is essentially the generalized Paley–Wiener theorem (Theorem 3.2), taking into account that, under the present assumptions, we have, by [3, Lemma 3], that φXcXc2 and so, by the consistency property for Mellin transform in Xc2, we obtain Mc[φ] = Mc[φ] = Mc2[φ].□

In conjunction with Proposition 3.3, Theorem 4.2 yields as a corollary that fHc1(Ha) dist(f(,0),Bσ1)=O(eaσ)(σ+),

a result contained in [8, Theorem 6].

In [7] we proved that functions from Mellin–Sobolev spaces have distances from Bc,σ1 that behave like 𝓞(σr). However, the familiar Mellin–Sobolev spaces are not appropriate for equivalences. The following modified space will (trivially) give the desired equivalence.

For r ∈ ℕ0 and α > 0, define Wcr+α,(R+):=fXc:supvR|vr[Θcαf]Mc(c+iv)|<+=fXc:supvR|vr+α[f]Mc(c+iv)|<+.

We note that Wcr+α,1(R+)Wcr+α,(R+). Now the following theorem is a simple consequence of [7, Theorem 5].

Theorem 4.3

(polynomial rate). Let fMc1. Then fWcr+α,(R+) dist(Θcαf,Bc,σ1)=O(σr)(σ+).

Proof

If fWcr+α,(R+) then there exists a constant C > 0 such that |vr[Θcαf]Mc(c+iv)|C

for every v ∈ ℝ. Hence, given σ > 0, we have |[Θcαf]Mc(c+iv)|Cσr

for every |v| ≥ σ, and so dist(Θcαf,Bc,σ1)=O(σr) as σ → + ∞.

Conversely, if the latter asymptotic equation holds, then |[Θcαf]Mc(c±iσ)|Cσr

for some constant C > 0 and every σ > 0, which implies that fWcr+α,(R+). □

Acknowledgements

Carlo Bardaro and Ilaria Mantellini have been partially supported by the “Gruppo Nazionale per l’Analisi Matematica e Applicazioni” (GNAMPA) of the “Istituto Nazionale di Alta Matematica” (INDAM) as well as by the Department of Mathematics and Computer Sciences of the University of Perugia.

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About the article

Received: 2017-07-07

Published Online: 2017-10-31

Published in Print: 2017-10-26


Citation Information: Fractional Calculus and Applied Analysis, Volume 20, Issue 5, Pages 1216–1238, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2017-0064.

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