An analog of Titchmarsh's theorem for the Laguerre–Bessel transform

Larbi Rakhimi (Department of Mathematics, Faculty of Sciences Ain Chock, University Hassan II Casablanca, Casablanca, Morocco)
Radouan Daher (Department of Mathematics, Faculty of Sciences Ain Chock, University Hassan II Casablanca, Casablanca, Morocco)

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Article publication date: 27 March 2023

548

Abstract

Purpose

Using a generalized translation operator, this study aims to obtain a generalization of Titchmarsh's theorem for the Laguerre–Bessel transform for functions satisfying the ψ-Laguerre–Bessel–Lipschitz condition in the space L2α (K), where K=0,+×0,+[.

Design/methodology/approach

The author has employed the results developed by Titchmarsh, of reference number [1].

Findings

In this paper, an analogous of Titchmarsh's theorem is established for Laguerre–Bessel transform.

Originality/value

To the best of the authors’ findings, at the time of submission of this paper, the results reported are new and interesting.

Keywords

Citation

Rakhimi, L. and Daher, R. (2023), "An analog of Titchmarsh's theorem for the Laguerre–Bessel transform", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-04-2022-0101

Publisher

:

Emerald Publishing Limited

Copyright © 2023, Larbi Rakhimi and Radouan Daher

License

Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode


1. Introduction

The integral Fourier transform, as Fourier series, is widely used in various fields of calculus, computational mathematics, mathematical physics, etc.

Years ago, Titchmarsh established ([1], Theorem 84) that if f satisfies the Lipschitz condition Lip(δ; p) in the Lp norm (1 < p ≤ 2) on the real line R, that is

Rf(x+h)f(x)p1p=o(hδ),(0<δ1)h0.

Then its Fourier transform F(f) belongs to Lβ(R), for

pp+δp1<βpp1.

A second result ([1], Theorem 85) characterized the set of functions in L2(R) satisfying the Cauchy–Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transform. Namely, we have:

Theorem 1.1.

If fL2(R). Then the following are equivalents:

  1. f(t+h)f(t)L2(R)=o(hδ),(0<δ<1) as h → 0.

  2. λrF(f)(λ)2dλ=o(r2δ) as r.

where F(f) stands for the Fourier transform of f.

Considerable attention has been devoted to discovering generalizations of new contexts for those theorems, see, e.g. ([2–7]). The aim of this paper is to give a generalization of these two theorems by using the harmonic analysis associated with the Laguerre–Bessel operators.

Throughout this paper, C denotes a positive constant which can differ from one line to another.

2. Preliminaries

Given α ≥ 0. The harmonic analysis on K=[0,+[×[0,+[ is generated by the following partial differential operators:

D1,α=2t2+2αttD2,α=2x2+2α+1xx+x2D1,α,
where (x,t)K. For (λ,m)[0,+[×N, the initial value problem:
D1,αu=λ2u,D2,αu=4λm+α+12uu(0,0)=1,ux(0,0)=ut(0,0)=0,
has a unique solution φλ,m given by
(1)φλ,m(x,t)=jα12(λt)Lmα(λx2),(x,t)K,
where Lmα is the Laguerre function defined on [0,+[, by
(2)Lmα(x)=ex2Lmα(x)Lmα(0),
Lmα being the Laguerre polynomial of degree m and order α, given by
(3)Lmα(x)=k=0m(1)kΓ(m+α+1)Γ(k+α+1)1k!(mk)!xk,
and jα is the normalized Bessel function given by
(4)jα(x)=Γ(α+1)k=0(1)kk!Γ(α+k+1)x22k.
Lemma 2.1.

[8] For all (λ,m)[0,+[×N, the function φλ,m is infinitely differentiable on R2, even with respect to each variable and we have

(5) sup(x,t)K|φλ,m(x,t)|=1.

Notation. We denote by:

  1. x,t=x,tK=x4+4t214 the homogeneous norm on K.

  2. λ,m=λ,m[0,+[×=4λm+α+12 the quasinorm on [0,+[×N. Let us denote Br, the ball centered 0 and of radius r, defined by,

    Br={(λ,m)[0,+[×N;λ,m<r}andBrc=([0,+[×N)\Br.

  3. Lαp(K),p[1,+], the spaces of measurable functions on K such that

    fp,α=K|f(x,t)|pdmα(x,t)1p<+,ifp[1,+[f,α=esssup(x,t)K|f(x,t)|<+,
    where dmα is the weighted Lebesgue measure on K, given by
    dmα(x,t)=x2α+1t2απΓ(α+1)dxdt.

  4. Lγαp([0,+[×N),p[1,+], the spaces of measurable functions on [0,+[×N such that

    gγα,p=[0,+[×N|g(λ,m)|pdγα(λ,m)1p<+, if p[1,+[gγα,=esssup(λ,m)[0,+[×N|g(λ,m)|<+,
    where α is the positive measure defined on [0,+[×N by
    [0,+[×Ng(λ,m)dγα(λ,m)=122α1Γα+12m=0Lmα(0)0+g(λ,m)λ3α+1dλ.

Definition 2.2.

  1. The translation operators T(x,t)(α),(x,t)K are defined for a continuous function f on K, by

    T(x,t)(α)f(y,s)=14πi,j=010πfΔθ(x,y),Y+(1)it+(1)jsdθ, if α=0bα[0,π]3fΔθ(x,y),Δθ(x,y)ξdμα(ξ,ψ,θ), if α>0.
    where Δθ(x,y)=x2+y2+2xycosθ,bα=(α+1)Γα+12π34Γ(α), Y = xy sin θ and
    dμα(ξ,ψ,θ)=(sinξ)2α1(sinψ)2α1(sinθ)2αdξdψdθ.

  2. The convolution product of two continuous functions f, g on K, with compact support is defined by

    (f*g)(x,t)=KT(x,t)(α)f(y,s)g(y,s)dmα(y,s),(x,t)K.

We have the following properties:

  1. If fLαp(K),gLαq(K) such that 1 ≤ p, q ≤  and 1p+1q1=1r, then the function f*gLαr(K), and

    f*gr,αfp,αgq,α.

  2. For all (λ,m)[0,+[×N, the kernel φλ,m verifies the following product formula

    φλ,m(x,t)φλ,m(y,s)=T(x,t)(α)φλ,m(y,s),(x,t),(y,s)K.

  3. For fLαp(K),p[1,+],we have T(x,t)(α)fLαp(K) and

    T(x,t)(α)fp,αfp,α.

The Fourier–Laguerre–Bessel transform of a function in Lα1(K) is given by

FLBf(λ,m)=Kf(x,t)φλ,m(x,t)dmα(x,t),(λ,m)[0,+[×N.

From Ref. [8], it is well known that Fourier–Laguerre–Bessel transform can be inverted to

FLB1f(x,t)=[0,+[×Nf(λ,m)φλ,m(x,t)dγα(λ,m),(x,t)K.

It is well-known (see Refs. [8–11]) that the Fourier–Laguerre–Bessel transform FLB satisfies the following properties.

Theorem 2.3.

(Inversion formula). If fLα1(K) such that FLB(f)Lγα1([0,+[×N), then for all (x,t)K we have

f(x,t)=[0,+[×NFLBf(λ,m)φλ,m(x,t)dγα(λ,m).

Theorem 2.4.

(Plancherel Theorem for FLB). The generalized Fourier transform FLB extends to an isometric isomorphism from Lα2(K). Onto Lγα2([0,+[×N).

Proposition 2.5.

For fLα1(K),(x,t)K and (λ,m)[0,+[×N, we have

(6) FLB(T(x,t)(α)f)(λ,m)=φλ,m(x,t)FLB(f)(λ,m).

Remark 1.

From (6) (see Ref. [14]), we get

(7) FLB(T(x,t)(α)ff)(λ,m)=(φλ,m(x,t)1)FLB(f)(λ,m).

3. Main results

In order to give the main results, we begin with auxiliary results interesting in themselves.

Lemma 3.1.

Let η > 0.

  1. The behavior in 0 of the kernel φλ,m could be expressed as follows:

    (8) φλ,m(x,t)=1(λt)24α+12λ,mx24(α+1)+κα,mλ2x4+o(λ,m2x,t4),
    where κα,m=m22(α+1)(α+2)+m2(α+2)+18.

  2. There exists a constant C, such that if |λ, m|x2 < η, then

    (9) |φλ,m(x,t)1|C|λ,m|x2.

  3. There exist C > 0 such that for all (x,t)K,

    (10) λ,m|x,t|2<ηφλ,m(x,t)12C|λ,m|2|x,t|4

  4. There exist C > 0 and A > 0 such that for all |x, t|2λ, m∣ > A and (x,t)K,

    (11) φλ,m(x,t)1C.

Proof.

  1. From the relation (2) and (3), we have

    (12)Lmα(x)=1m+α+12α+1x+m22(α+1)(α+2)+m2(α+2)+18x2+o(x2).

Then (i) could be deduced easily using the relation (1),(12) and the behavior in 0 of the normalized Bessel function which states

jα(u)=114(α+1)u2+o(u2).
  1. Using relations (8), we obtain

    lim|λ,m|x20|φλ,m(x,t)1||λ,m|x2=14(α+1)>0,
    which proves the wanted result.

  2. Using relation (8).

  3. From ([6], Lemma 4.3), we have

    limλ,m+ψλ,m(x,t)=0,
    where ψλ,m(x,t)=eiλtLmα(λx2) the Laguerre kernel, and from Ref. [12], we have the asymptotic formula for the normalized Bessel function jα when x → + :
    jα(x)=Γ(α+1)Γ122xα+12cosx(2α+1)π4+o1x32.

Hence as

φλ,m(x,t)=jα12(λt)1eiλtψλ,m(x,t),
then limλ,m∣→+φλ,m(x, t) = 0, we get limλ,m∣→+φλ,m(x, t) − 1∣ = 1, which completes the proof. □
Lemma 3.2.

(Hausdorff-Young inequality) Let 1 < p ≤ 2. If fLαp(K), then FLBfLγαq([0,+[×N) and we have

FLBfγα,qCfp,α,
where the numbers p and q above are conjugate exponents:
1p+1q=1.

Proof. By applying the Riesz–Thorin interpolation theorem to the elementary estimate [13] and Plancherel theorem, we obtain the desired inequality. □

Proposition 3.3.

Let f be a function in Lαp(K), such that T(x,t)(α)ffp,α=Oxγ for 1 < p ≤ 2 and 0 < γ ≤ 1. Then FLBf belongs to Lγαβ([0,+[×N), where

(α+2)p(α+2)(p1)+γp2<βpp1.

Proof. By proceeding similarly to theorem (Theorem 3.1 [6]). For fixed (x,t)K, we have using relations (7) and Lemma 3.2

[0,+[×Nφλ,m(x,t)1qFLBf(λ,m)qdγα(λ,m)=Oxγq.

Using relations (9), we get

Bηx2|λ,m|qFLBf(λ,m)qdγαx2qBηx2φλ,m(x,t)1qFLBf(λ,m)qdγαCx(γ2)q.

Now, let β ≤ q. From Hölder inequality, one gets

BX|λ,m|βFLBf(λ,m)βdγαBX|λ,m|qFLBf(λ,m)qdγαβqBX1dγα1βq.

Therefore

(13)BX|λ,m|βFLBf(λ,m)βdγα(λ,m)=OX(2γ)q2βq+(α+2)1βq.

Recall that B1c=([0,+[×N)\B1. To get the theorem, it is enough to prove that B1cBXFLBf(λ,m)βdγα(λ,m) is bounded when X → +. Therefore, we can write

B1cBXFLBf(λ,m)βdγα(λ,m)=122α1Γα+12m=0+Lmα(0)I,
where I depend on m and X and has the expression
I=14m+2a+2X4m+2α+2FLBf(λ,m)βλ3α+1dλ.

Then

I=14m+2α+2X4m+2α+24m+2α+2|λ,m|βΦm'|λ,m|dλ,
where
ΦmX=1xm+2a+2X2m+2a+2|λ,m|βFLBfλ,mβλ3α+1dλ

Making a change of variables and an integration by parts, we get

I=Φm(X)Xβ+β1Xtβ1Φm(t)dt.

Consequently

B1cBXFLBf(λ,m)βdγα(λ,m)=Xβψ(X)+β1Xtβ1ψ(t)dt,
where
ψX=122α1Γα+12m=0+Lmα0ΦmX,=B1cBX|λ,m|βFLBfλ,mβdγα(λ,m).

From relation (13), we have

B1cBXFLBf(λ,m)βdγα(λ,m)=OXβ+2γ2β+(α+2)1βq+O1Xtβ1t2γ2β+(α+2)1βqdt.

This is bounded as X → + if βγ2+α+2q+(α+2)<0 that gives β>(α+2)p(α+2)(p1)+γp2.□

Next we define the ψ-Laguerre–Bessel–Lipschitz class:

Definition 3.4.

A function f is said to be in ψ-Laguerre–Bessel–Lipschitz class and is denoted by Lipα(ψ, 2), if f belongs to Lα2(K) and verifies, for all (x,t)K

(14) T(x,t)(α)ff2,α=Oψ(|x,t|)as|x,t|0,
where
  1. ψ(t) is a continuous increasing function on [0;[.

  2. ψ(0) = 0 and ψ(ts) = ψ(t)ψ(s) for all t,s[0;[.

  3. 01hsψs1ds=o1h2ψh as  h0.

Example 3.5.

Let ψ(t) = tγ, where 0 < γ < 1. In this case, the relation (14) is a generalization of Lipschitz condition f(x+h)f(x)=Ohγ, and the ψ-Laguerre–Bessel–Lipschitz class Lipα(ψ, 2), are called the Laguerre–Bessel–Lipschitz class Lipα(γ, 2).

Now, we are able to generalise the equivalence theorem.

Theorem 3.6.

Let fLα2(K), the following two conditions are equivalent:

  1. f ∈ Lipα(ψ, 2).

  2. BrcFLBfλ,m2dγαλ,m=oψr1  as  r+.

Proof. (iii): Let fLα2(K) from (i), we have

T(x,t)(α)ff2,α2=[0,+[×Nφλ,m(x,t)12FLBf(λ,m)2dγα.

Therefore, using relation (11), we have

BA|x,t|2cFLBf(λ,m)2dγαCBA|x,t|2cφλ,m(x,t)12FLBf(λ,m)2dγαCT(x,t)(α)ff2,α2=Oψ(|x,t|2).

Consequently, (ii) holds.

(iii): Denote r=η|x,t|2, by Plancherel theorem, we get

T(x,t)(α)ff2,α2=I1+I2,
where
I1=Brφλ,m(x,t)12FLBf(λ,m)2dγα(λ,m)
and
I2=Brcφλ,m(x,t)12FLBf(λ,m)2dγα(λ,m).

Using relation (5), we find that

I24BrcFLBf(λ,m)2dγα=o(ψ(r1))=o(ψ(|x,t|2)).

Denote

g(X)=XFLBf(λ,m)2λ3α+1dλ,
then g(λ)=FLBf(λ,m)2λ3α+1. Using relation (10), which gives
I1C|x,t|4122α1Γα+12m=0Lmα(0)0η4κm|x,t|24κm2λ2g(λ)dλ,
by integration by parts, we have
I1C|x,t|4122α1Γα+12m=0Lmα(0)η2|x,t|4gη4κm|x,t|2+4κm20η4κm|x,t|22λg(λ)dλ.

Remark that

122α1Γα+12m=0Lmα(0)gR4κm=BRcFLBf(λ,m)2dγα=o(ψ(R1)).

Making a change of variable, one gets

I1o(ψ(|x,t|2))+C|x,t|40η|x,t|2u122α1Γα+12m=0Lmα(0)gu4κmdu=o(ψ(|x,t|2))+x,t4o0ηx,t2uψ(u1)du=o(ψ(|x,t|2)).

Then

T(x,t)(α)ff2,α2=o(ψ(|x,t|2))=o(ψ(|x,t|)2) as |x,t|0.

We conclude this work by the following immediate consequence. It is analogous of Titchmarsh theorem ([1], Theorem 85) is established for Laguerre–Bessel transform.

Corollary 3.7.

Let ψ(t) = tγ, where 0 < γ < 1. The following two conditions are equivalent:

  1. f is in LaguerreBesselLipschitz class Lipα(γ, 2).

  2. BrcFLBf(λ,m)2dγα(λ,m)=Orγ as r → +.

The authors would like to thank the referee for carefully checking the manuscript and for his/her helpful comments and suggestions.

Data Availability Statement: The manuscript has no associated data.

References

1.Titchmarsh EC. Introduction to the theory of fourier integral. Amen House, London. E. C. 4: Oxford University Press; 1948.

2.El Ouadih S, Daher R. Characterization of Dini-Lipschitz functions for the Helgason Fourier transform on rank one symmetric spaces. Adv Pure Appl Math. 2016; 7(4): 223-30.

3.Daher R, El Hamma M. An analog of Titchmarsh's theorem for the generalized Dunkl transform. J.Pseud Diff Op Appl. 2016; 7(1): 59-65.

4.Younis MS. Fourier transform of Lipschitz functions on compact groups. Ph.D. Thesis. Hamilton, ON, Canada: McMaster University; 1974.

5.El ouadih S, Daher R. Generalization of Titchmarsh's theorem for the Dunkl transform in the space Lp(Rd;wl(x)dx).

6.Negzaoui S. Lipschitz conditions in Laguerre hypergroup. Mediterr J Math. 2017; 14: 191. doi: 10.1007/s00009-017-0989-4.

7.Achak A, Bouhlal A, Daher R. Titchmarsh's theorem and some remarks concerning the right-sided quaternion Fourier transform. Bol Soc Mat Mex. 2020; 26: 599-616.

8.Jebbari E, Sifi M, Soltani F. Laguerre-Bessel wavelet transform. Glob J Pure Appl. Math. 2005; 1: 13-26.

9.Kortas H, Sifi M. Levy-Khintchine formula and dual convolution semigroups associated with Laguerre and Bessel functions. Potential Anal. 2001; 15: 43-58.

10.Jebbari E. Harmonic analysis associated with system of partial differential operators D1 and D2, Preprint, 2004.

11.Soumeya H, Lotfi K. Uncertainty principle inequalities related to Laguerre-Bessel transform.

12.Watson GN. A treatise on the theory of Bessel functions. Cambridge: Cambridge University Press; 1966.

13.Folland G. Real analysis modern techniques and their applications, Pure Appl. Math. 2nd ed. New York: Wiley; 1999.

14Rakhimi Larbi, Daher Radouan. Modulus of Smoothness and K-Functionals Constructed by Generalized Laguerre-Bessel Operator. Tatra Mountains Mathematical Publications. 2022; 81(1). doi: 10.2478/tmmp-2022-0008.

Corresponding author

Larbi Rakhimi can be contacted at: larbi.rakimi-etu@etu.univh2c.ma

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