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

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 L²_α (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.

Citation

Rakhimi, L. and Daher, R. (2024), "An analog of Titchmarsh's theorem for the Laguerre–Bessel transform", Arab Journal of Mathematical Sciences, Vol. 30 No. 2, pp. 235-244. https://doi.org/10.1108/AJMS-04-2022-0101

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 L^p norm (1 < p ≤ 2) on the real line R, that is

∫R∣f(x+h)−f(x)∣p1p=o(hδ),(0<δ≤1)h→0.

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

pp+δp−1<β≤pp−1.

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 f∈L2(R). Then the following are equivalents:

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

2. ∫∣λ∣≥r∣F(f)(λ)∣2dλ=o(r−2δ) 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=∂2∂t2+2αt∂∂tD2,α=∂2∂x2+2α+1x∂∂x+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,∂u∂x(0,0)=∂u∂t(0,0)=0,

has a unique solution φ_λ,m given by

(1)φλ,m(x,t)=jα−12(λt)Lmα(λx2),(x,t)∈K,

where Lmα is the Laguerre function defined on [0,+∞[, by

(2)Lmα(x)=e−x2Lmα(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!(m−k)!xk,

and j_α 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 φ_λ,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,t∣K=x4+4t214 the homogeneous norm on K.

2. ∣λ,m∣=∣λ,m∣[0,+∞[×ℕ=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

   ‖f‖p,α=∫K|f(x,t)|pdmα(x,t)1p<+∞,ifp∈[1,+∞[‖f‖∞,α=esssup(x,t)∈K|f(x,t)|<+∞,

   where dm_α 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

   ‖g‖γα,p=∫[0,+∞[×N|g(λ,m)|pdγα(λ,m)1p<+∞, if p∈[1,+∞[‖g‖γα,∞=esssup(λ,m)∈[0,+∞[×N|g(λ,m)|<+∞,

   where dγ_α is the positive measure defined on [0,+∞[×N by

   ∫[0,+∞[×Ng(λ,m)dγα(λ,m)=122α−1Γα+12∑m=0∞Lmα(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)=14π∑i,j=01∫0πfΔθ(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+2xy⁡cos⁡θ,bα=(α+1)Γα+12π34Γ(α), Y = xy sin θ and

   dμα(ξ,ψ,θ)=(sin⁡ξ)2α−1(sin⁡ψ)2α−1(sin⁡θ)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 f∈Lαp(K),g∈Lαq(K) such that 1 ≤ p, q ≤ ∞ and 1p+1q−1=1r, then the function f*g∈Lαr(K), and

   ‖f*g‖r,α≤‖f‖p,α‖g‖q,α.

2. For all (λ,m)∈[0,+∞[×N, the kernel φ_λ,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 f∈Lαp(K),p∈[1,+∞],we have T(x,t)(α)f∈Lαp(K) and

   ‖T(x,t)(α)f‖p,α≤‖f‖p,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

FLB−1f(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 f∈Lα1(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 f∈Lα1(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)f−f)(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 φ_λ,m could be expressed as follows:

   (8) φλ,m(x,t)=1−(λt)24α+12−∣λ,m∣x24(α+1)+κα,mλ2x4+o(∣λ,m∣2∣x,t∣4),

   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<η⇒φλ,m(x,t)−12≤C|λ,m|2|x,t|4

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

   (11) ∣φλ,m(x,t)−1∣≥C.

Proof.

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

   (12)Lmα(x)=1−m+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)=1−14(α+1)u2+o(u2).

1. Using relations (8), we obtain

   lim|λ,m|x2→0|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

   lim∣λ,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 j_α when x → + ∞:

   jα(x)=Γ(α+1)Γ122xα+12⁡cosx−(2α+1)π4+o1x32.

Hence as

φλ,m(x,t)=jα−12(λt)1eiλtψλ,m(x,t),

then lim_{∣λ,m∣→+∞}φ_λ,m(x, t) = 0, we get lim_{∣λ,m∣→+∞}∣φ_λ,m(x, t) − 1∣ = 1, which completes the proof. □

Lemma 3.2.

(Hausdorff-Young inequality) Let 1 < p ≤ 2. If f∈Lαp(K), then FLBf∈Lγαq([0,+∞[×N) and we have

‖FLBf‖γα,q≤C‖f‖p,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)f−fp,α=Oxγ for 1 < p ≤ 2 and 0 < γ ≤ 1. Then FLBf belongs to Lγαβ([0,+∞[×N), where

(α+2)p(α+2)(p−1)+γp2<β≤pp−1.

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)qdγα≤x−2q∫Bηx2φλ,m(x,t)−1qFLBf(λ,m)qdγα≤Cx(γ−2)q.

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

∫BX|λ,m|βFLBf(λ,m)βdγα≤∫BX|λ,m|qFLBf(λ,m)qdγαβq∫BX1dγα1−βq.

Therefore

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

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

∫B1c∩BXFLBf(λ,m)βdγα(λ,m)=122α−1Γα+12∑m=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)X−β+β∫1Xt−β−1Φm(t)dt.

Consequently

∫B1c∩BXFLBf(λ,m)βdγα(λ,m)=X−βψ(X)+β∫1Xt−β−1ψ(t)dt,

where

ψX=122α−1Γα+12∑m=0+∞Lmα0ΦmX,=∫B1c∩BX|λ,m|βFLBfλ,mβdγα(λ,m).

From relation (13), we have

∫B1c∩BXFLBf(λ,m)βdγα(λ,m)=OX−β+2−γ2β+(α+2)1−βq+O∫1Xt−β−1t2−γ2β+(α+2)1−βqdt.

This is bounded as X → +∞ if −βγ2+α+2q+(α+2)<0 that gives β>(α+2)p(α+2)(p−1)+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 Lip_α(ψ, 2), if f belongs to Lα2(K) and verifies, for all (x,t)∈K

(14) T(x,t)(α)f−f2,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. ∫01hsψs−1ds=o1h2ψh as  h→0.

Example 3.5.

Let ψ(t) = t^γ, 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 Lip_α(ψ, 2), are called the Laguerre–Bessel–Lipschitz class Lip_α(γ, 2).

Now, we are able to generalise the equivalence theorem.

Theorem 3.6.

Let f∈Lα2(K), the following two conditions are equivalent:

1. f ∈ Lip_α(ψ, 2).

2. ∫BrcFLBfλ,m2dγαλ,m=oψr−1  as  r→+∞.

Proof. (i ⇒ ii): Let f∈Lα2(K) from (i), we have

T(x,t)(α)f−f2,α2=∫[0,+∞[×Nφλ,m(x,t)−12FLBf(λ,m)2dγα.

Therefore, using relation (11), we have

∫BA|x,t|2cFLBf(λ,m)2dγα≤C∫BA|x,t|2cφλ,m(x,t)−12FLBf(λ,m)2dγα≤CT(x,t)(α)f−f2,α2=Oψ(|x,t|2).

Consequently, (ii) holds.

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

T(x,t)(α)f−f2,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

I2≤4∫BrcFLBf(λ,m)2dγα=o(ψ(r−1))=o(ψ(|x,t|2)).

Denote

g(X)=∫X∞FLBf(λ,m)2λ3α+1dλ,

then g′(λ)=−FLBf(λ,m)2λ3α+1. Using relation (10), which gives

I1≤C|x,t|4122α−1Γα+12∑m=0∞Lmα(0)∫0η4κm|x,t|24κm2λ2−g′(λ)dλ,

by integration by parts, we have

I1≤C|x,t|4122α−1Γα+12∑m=0∞Lmα(0)−η2|x,t|4gη4κm|x,t|2+4κm2∫0η4κm|x,t|22λg(λ)dλ.

Remark that

122α−1Γα+12∑m=0∞Lmα(0)gR4κm=∫BRcFLBf(λ,m)2dγα=o(ψ(R−1)).

Making a change of variable, one gets

I1≤o(ψ(|x,t|2))+C|x,t|4∫0η|x,t|2u122α−1Γα+12∑m=0∞Lmα(0)gu4κmdu=o(ψ(|x,t|2))+∣x,t∣4o∫0η∣x,t∣2uψ(u−1)du=o(ψ(|x,t|2)).

Then

T(x,t)(α)f−f2,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) = t^γ, where 0 < γ < 1. The following two conditions are equivalent:

1. f is in Laguerre–Bessel–Lipschitz class Lip_α(γ, 2).

2. ∫BrcFLBf(λ,m)2dγα(λ,m)=Or−γ 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.