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

Rakhimi, Larbi; Daher, Radouan

doi:10.1108/AJMS-04-2022-0101

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 $K = [0, + \infty [\times [0, + \infty [$ ⁠.

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.

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

{(\int_{R} ∣ f (x + h) - f (x) ∣^{p})}^{\frac{1}{p}} = o (h^{δ}), (0 < δ \leq 1) h \to 0 .

Then its Fourier transform $F (f)$ belongs to $L^{β} (R)$ ⁠, for

\frac{p}{p + δ p - 1} < β \leq \frac{p}{p - 1} .

A second result ([1], Theorem 85) characterized the set of functions in $L^{2} (R)$ 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 $f \in L^{2} (R)$ ⁠. Then the following are equivalents:

$‖ f (t + h) - f (t) ‖_{L^{2} (R)} = o (h^{δ}), (0 < δ < 1)$ as h → 0.
$\int_{∣ λ ∣ \geq r} ∣ F (f) (λ) ∣^{2} d λ = o (r^{- 2 δ})$ 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 $K = [0, + \infty [\times [0, + \infty [$ is generated by the following partial differential operators:

\{\begin{cases} D_{1, α} = \frac{\partial^{2}}{\partial t^{2}} + \frac{2 α}{t} \frac{\partial}{\partial t} \\ D_{2, α} = \frac{\partial^{2}}{\partial x^{2}} + \frac{2 α + 1}{x} \frac{\partial}{\partial x} + x^{2} D_{1, α}, \end{cases}

where $(x, t) \in K$ ⁠. For $(λ, m) \in [0, + \infty [\times N$ ⁠, the initial value problem:

\{\begin{cases} D_{1, α} u = - λ^{2} u, \\ D_{2, α} u = - 4 λ (m + \frac{α + 1}{2}) u \\ u (0,0) = 1, \frac{\partial u}{\partial x} (0,0) = \frac{\partial u}{\partial t} (0,0) = 0, \end{cases}

has a unique solution φ_λ,m given by

φ_{λ, m} (x, t) = j_{α - \frac{1}{2}} (λ t) L_{m}^{α} (λ x^{2}), (x, t) \in K,

(1)

where $L_{m}^{α}$ is the Laguerre function defined on $[0, + \infty [$ ⁠, by

L_{m}^{α} (x) = e^{- \frac{x}{2}} \frac{L_{m}^{α} (x)}{L_{m}^{α} (0)},

(2)

$L_{m}^{α}$ being the Laguerre polynomial of degree m and order α, given by

L_{m}^{α} (x) = \sum_{k = 0}^{m} {(- 1)}^{k} \frac{Γ (m + α + 1)}{Γ (k + α + 1)} \frac{1}{k! (m - k)!} x^{k},

(3)

and j_α is the normalized Bessel function given by

j_{α} (x) = Γ (α + 1) \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{k! Γ (α + k + 1)} {(\frac{x}{2})}^{2 k} .

(4)

Lemma 2.1.

[8] For all $(λ, m) \in [0, + \infty [\times N$ , the function φ_λ,m is infinitely differentiable on $R^{2}$ , even with respect to each variable and we have

s u p_{(x, t) \in K} | φ_{λ, m} (x, t) | = 1 .

(5)

Notation. We denote by:

$∣ x, t ∣ = ∣ (x, t) ∣_{K} = {(x^{4} + 4 t^{2})}^{\frac{1}{4}}$ the homogeneous norm on $K$ ⁠.
$∣ λ, m ∣ = ∣ (λ, m) ∣_{[0, + \infty [\times ℕ} = 4 λ (m + \frac{α + 1}{2})$ the quasinorm on $[0, + \infty [\times N$ ⁠. Let us denote $B_{r}$ ⁠, the ball centered 0 and of radius r, defined by,
$B_{r} = {(λ, m) \in [0, + \infty [\times N; ∣ λ, m ∣ < r} a n d B_{r}^{c} = ([0, + \infty [\times N) \ B_{r} .$
$L_{α}^{p} (K), p \in [1, + \infty]$ ⁠, the spaces of measurable functions on $K$ such that
$\{\begin{cases} ‖ f ‖_{p, α} = {[\int_{K} | f (x, t) |^{p} d m_{α} (x, t)]}^{\frac{1}{p}} < + \infty, i f p \in [1, + \infty [ \\ ‖ f ‖_{\infty, α} = e s s \sup_{(x, t) \in K} | f (x, t) | < + \infty, \end{cases}$
where dm_α is the weighted Lebesgue measure on $K$ ⁠, given by
$d m_{α} (x, t) = \frac{x^{2 α + 1} t^{2 α}}{π Γ (α + 1)} d x d t .$
$L_{γ_{α}}^{p} ([0, + \infty [\times N), p \in [1, + \infty]$ ⁠, the spaces of measurable functions on $[0, + \infty [\times N$ such that
$\{\begin{cases} ‖ g ‖_{γ_{α, p}} = {[\int_{[0, + \infty [\times N} | g (λ, m) |^{p} d γ_{α} (λ, m)]}^{\frac{1}{p}} < + \infty, if p \in [1, + \infty [ \\ ‖ g ‖_{γ_{α, \infty}} = e s s \sup_{(λ, m) \in [0, + \infty [\times N} | g (λ, m) | < + \infty, \end{cases}$
where dγ_α is the positive measure defined on $[0, + \infty [\times N$ by
$\int_{[0, + \infty [\times N} g (λ, m) d γ_{α} (λ, m) = \frac{1}{2^{2 α - 1} Γ (α + \frac{1}{2})} \sum_{m = 0}^{\infty} L_{m}^{α} (0) \int_{0}^{+ \infty} g (λ, m) λ^{3 α + 1} d λ .$

Definition 2.2.

The translation operators $T_{(x, t)}^{(α)}, (x, t) \in K$ are defined for a continuous function f on $K$ , by
$T_{(x, t)}^{(α)} f (y, s) = \{\begin{cases} \frac{1}{4 π} \sum_{i, j = 0}^{1} \int_{0}^{π} f (Δ_{θ} (x, y), Y + {(- 1)}^{i} t + {(- 1)}^{j} s) d θ, if α = 0 \\ b_{α} \int_{{[0, π]}^{3}} f (Δ_{θ} (x, y), Δ_{θ} (x, y) ξ) d μ_{α} (ξ, ψ, θ), if α > 0 . \end{cases}$
where $Δ_{θ} (x, y) = \sqrt{x^{2} + y^{2} + 2 x y \cos θ}, b_{α} = \frac{(α + 1) Γ (α + \frac{1}{2})}{π^{\frac{3}{4}} Γ (α)}$ ,Y = xy sin θand
$d μ_{α} (ξ, ψ, θ) = {(\sin ξ)}^{2 α - 1} {(\sin ψ)}^{2 α - 1} {(\sin θ)}^{2 α} d ξ d ψ d θ .$
The convolution product of two continuous functions f, g on $K$ , with compact support is defined by
$(f * g) (x, t) = \int_{K} T_{(x, t)}^{(α)} f (y, s) g (y, s) d m_{α} (y, s), (x, t) \in K .$

We have the following properties:

If $f \in L_{α}^{p} (K), g \in L_{α}^{q} (K)$ such that 1 ≤ p, q ≤ ∞ and $\frac{1}{p} + \frac{1}{q} - 1 = \frac{1}{r}$ ⁠, then the function $f * g \in L_{α}^{r} (K)$ ⁠, and
$‖ f * g ‖_{r, α} \leq ‖ f ‖_{p, α} ‖ g ‖_{q, α} .$
For all $(λ, m) \in [0, + \infty [\times N$ ⁠, the kernel φ_λ,m verifies the following product formula
$φ_{λ, m} (x, t) φ_{λ, m} (y, s) = T_{(x, t)}^{(α)} φ_{λ, m} (y, s), (x, t), (y, s) \in K .$
For $f \in L_{α}^{p} (K), p \in [1, + \infty]$ ⁠,we have $T_{(x, t)}^{(α)} f \in L_{α}^{p} (K)$ and
$‖ T_{(x, t)}^{(α)} f ‖_{p, α} \leq ‖ f ‖_{p, α} .$

The Fourier–Laguerre–Bessel transform of a function in $L_{α}^{1} (K)$ is given by

F_{L B} f (λ, m) = \int_{K} f (x, t) φ_{λ, m} (x, t) d m_{α} (x, t), (λ, m) \in [0, + \infty [\times N .

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

F_{L B}^{- 1} f (x, t) = \int_{[0, + \infty [\times N} f (λ, m) φ_{λ, m} (x, t) d γ_{α} (λ, m), (x, t) \in K .

It is well-known (see Refs. [8–11]) that the Fourier–Laguerre–Bessel transform $F_{L B}$ satisfies the following properties.

Theorem 2.3.

(Inversion formula). If $f \in L_{α}^{1} (K)$ such that $F_{L B} (f) \in L_{γ_{α}}^{1} ([0, + \infty [\times N)$ , then for all $(x, t) \in K$ we have

f (x, t) = \int_{[0, + \infty [\times N} F_{L B} f (λ, m) φ_{λ, m} (x, t) d γ_{α} (λ, m) .

Theorem 2.4.

(Plancherel Theorem for $F_{L B}$ ). The generalized Fourier transform $F_{L B}$ extends to an isometric isomorphism from $L_{α}^{2} (K)$ ⁠. Onto $L_{γ_{α}}^{2} ([0, + \infty [\times N)$ ⁠.

Proposition 2.5.

For $f \in L_{α}^{1} (K), (x, t) \in K$ and $(λ, m) \in [0, + \infty [\times N$ , we have

F_{L B} (T_{(x, t)}^{(α)} f) (λ, m) = φ_{λ, m} (x, t) F_{L B} (f) (λ, m) .

(6)

Remark 1.

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

F_{L B} (T_{(x, t)}^{(α)} f - f) (λ, m) = (φ_{λ, m} (x, t) - 1) F_{L B} (f) (λ, m) .

(7)

3. Main results

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

Lemma 3.1.

Let η > 0.

The behavior in 0 of the kernel φ_λ,m could be expressed as follows:
$φ_{λ, m} (x, t) = 1 - \frac{{(λ t)}^{2}}{4 (α + \frac{1}{2})} - \frac{∣ λ, m ∣ x^{2}}{4 (α + 1)} + κ_{α, m} λ^{2} x^{4} + o (∣ λ, m ∣^{2} ∣ x, t ∣^{4}),$
(8)
where $κ_{α, m} = \frac{m^{2}}{2 (α + 1) (α + 2)} + \frac{m}{2 (α + 2)} + \frac{1}{8}$ ⁠.
There exists a constant C, such that if |λ, m|x² < η, then
$| φ_{λ, m} (x, t) - 1 | \geq C | λ, m | x^{2} .$
(9)
There exist C > 0 such that for all $(x, t) \in K$ ,
$∣ λ, m ∣ | x, t |^{2} < η \Rightarrow {|φ_{λ, m} (x, t) - 1|}^{2} \leq C | λ, m |^{2} | x, t |^{4}$
(10)
There exist C > 0 and A > 0 such that for all |x, t|²∣λ, m∣ > A and $(x, t) \in K$ ⁠,
$∣ φ_{λ, m} (x, t) - 1 ∣ \geq C .$
(11)

Proof.

From the relation (2) and (3), we have
$L_{m}^{α} (x) = 1 - \frac{m + \frac{α + 1}{2}}{α + 1} x + (\frac{m^{2}}{2 (α + 1) (α + 2)} + \frac{m}{2 (α + 2)} + \frac{1}{8}) x^{2} + o (x^{2}) .$
(12)

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

j_{α} (u) = 1 - \frac{1}{4 (α + 1)} u^{2} + o (u^{2}) .

Using relations (8), we obtain
$\lim_{| λ, m | x^{2} \to 0} (\frac{| φ_{λ, m} (x, t) - 1 |}{| λ, m | x^{2}}) = \frac{1}{4 (α + 1)} > 0,$
which proves the wanted result.
Using relation (8).
From ([6], Lemma 4.3), we have
$\lim_{∣ λ, m ∣ \to + \infty} ψ_{λ, m} (x, t) = 0,$
where $ψ_{λ, m} (x, t) = e^{i λ t} L_{m}^{α} (λ x^{2})$ the Laguerre kernel, and from Ref. [12], we have the asymptotic formula for the normalized Bessel function j_α when x → + ∞:
$j_{α} (x) = \frac{Γ (α + 1)}{Γ (\frac{1}{2})} {(\frac{2}{x})}^{α + \frac{1}{2}} \cos (x - (2 α + 1) \frac{π}{4}) + o (\frac{1}{x^{\frac{3}{2}}}) .$

Hence as

φ_{λ, m} (x, t) = j_{α - \frac{1}{2}} (λ t) \frac{1}{e^{i λ t}} ψ_{λ, 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 $f \in L_{α}^{p} (K)$ , then $F_{L B} f \in L_{γ_{α}}^{q} ([0, + \infty [\times N)$ and we have

‖ F_{L B} f ‖_{γ_{α, q}} \leq C ‖ f ‖_{p, α},

where the numberspandqabove are conjugate exponents:

\frac{1}{p} + \frac{1}{q} = 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 $L_{α}^{p} (K)$ , such that ${‖T_{(x, t)}^{(α)} f - f‖}_{p, α} = O (x^{γ})$ for 1 < p ≤ 2 and 0 < γ ≤ 1. Then $F_{L B} f$ belongs to $L_{γ_{α}}^{β} ([0, + \infty [\times N)$ , where

\frac{(α + 2) p}{(α + 2) (p - 1) + \frac{γ p}{2}} < β \leq \frac{p}{p - 1} .

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

\int_{[0, + \infty [\times N} {|φ_{λ, m} (x, t) - 1|}^{q} {|F_{L B} f (λ, m)|}^{q} d γ_{α} (λ, m) = O (x^{γ q}) .

Using relations (9), we get

\begin{aligned} \int_{B_{\frac{η}{x^{2}}}} | λ, m |^{q} {|F_{L B} f (λ, m)|}^{q} d γ_{α} & \leq x^{- 2 q} \int_{B_{\frac{η}{x^{2}}}} {|φ_{λ, m} (x, t) - 1|}^{q} {|F_{L B} f (λ, m)|}^{q} d γ_{α} \\ \leq C x^{(γ - 2) q} . \end{aligned}

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

\int_{B_{X}} | λ, m |^{β} {|F_{L B} f (λ, m)|}^{β} d γ_{α} \leq {(\int_{B_{X}} | λ, m |^{q} {|F_{L B} f (λ, m)|}^{q} d γ_{α})}^{\frac{β}{q}} {(\int_{B_{X}} 1 d γ_{α})}^{1 - \frac{β}{q}} .

Therefore

\int_{B_{X}} | λ, m |^{β} {|F_{L B} f (λ, m)|}^{β} d γ_{α} (λ, m) = O (X^{\frac{(2 - γ) q}{2} \frac{β}{q} + (α + 2) (1 - \frac{β}{q})}) .

(13)

Recall that $B_{1}^{c} = ([0, + \infty [\times N) \ B_{1}$ ⁠. To get the theorem, it is enough to prove that $\int_{B_{1}^{c} \cap B_{X}} {|F_{L B} f (λ, m)|}^{β} d γ_{α} (λ, m)$ is bounded when X → +∞. Therefore, we can write

\int_{B_{1}^{c} \cap B_{X}} {|F_{L B} f (λ, m)|}^{β} d γ_{α} (λ, m) = \frac{1}{2^{2 α - 1} Γ (α + \frac{1}{2})} \sum_{m = 0}^{+ \infty} L_{m}^{α} (0) I,

where I depend on m and X and has the expression

I = \int_{\frac{1}{4 m + 2 a + 2}}^{\frac{X}{4 m + 2 α + 2}} {|F_{L B} f (λ, m)|}^{β} λ^{3 α + 1} d λ .

Then

I = \int_{\frac{1}{4 m + 2 α + 2}}^{\frac{X}{4 m + 2 α + 2}} (4 m + 2 α + 2) | λ, m |^{- β} Φ_{m}^{'} (| λ, m |) d λ,

where

Φ_{m} (X) = \int_{\frac{1}{x m + 2 a + 2}}^{\frac{X}{2 m + 2 a + 2}} | λ, m |^{β} {|F_{L B} f (λ, m)|}^{β} λ^{3 α + 1} d λ

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

I = Φ_{m} (X) X^{- β} + β \int_{1}^{X} t^{- β - 1} Φ_{m} (t) d t .

Consequently

\int_{B_{1}^{c} \cap B_{X}} {|F_{L B} f (λ, m)|}^{β} d γ_{α} (λ, m) = X^{- β} ψ (X) + β \int_{1}^{X} t^{- β - 1} ψ (t) d t,

where

\begin{matrix} ψ (X) & = & \frac{1}{2^{2 α - 1} Γ (α + \frac{1}{2})} \sum_{m = 0}^{+ \infty} L_{m}^{α} (0) Φ_{m} (X), \\ = & \int_{B_{1}^{c} \cap B_{X}} | λ, m |^{β} {|F_{L B} f (λ, m)|}^{β} d γ_{α} (λ, m) . \end{matrix}

From relation (13), we have

\begin{aligned} \int_{B_{1}^{c} \cap B_{X}} {|F_{L B} f (λ, m)|}^{β} d γ_{α} (λ, m) = & O (X^{- β + \frac{2 - γ}{2} β + (α + 2) (1 - \frac{β}{q})}) \\ + O (\int_{1}^{X} t^{- β - 1} t^{\frac{2 - γ}{2} β + (α + 2) (1 - \frac{β}{q})} d t) . \end{aligned}

This is bounded as X → +∞ if $- β (\frac{γ}{2} + \frac{α + 2}{q}) + (α + 2) < 0$ that gives $β > \frac{(α + 2) p}{(α + 2) (p - 1) + \frac{γ p}{2}}$ ⁠.□

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 $L_{α}^{2} (K)$ and verifies, for all $(x, t) \in K$

{‖T_{(x, t)}^{(α)} f - f‖}_{2, α} = O (ψ (| x, t |)) a s | x, t | \to 0,

(14)

where

ψ(t) is a continuous increasing function on $[0; \infty [$ ⁠.
ψ(0) = 0 and ψ(ts) = ψ(t)ψ(s) for all $t, s \in [0; \infty [$ ⁠.
$\int_{0}^{\frac{1}{h}} s ψ (s^{- 1}) d s = o (\frac{1}{h^{2}} ψ (h)) a s h \to 0$ ⁠.

Example 3.5.

Let ψ(t) = t^γ, where 0 < γ < 1. In this case, the relation (14) is a generalization of Lipschitz condition $‖ f (x + h) - f (x) ‖ = O (h^{γ})$ ⁠, 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 $f \in L_{α}^{2} (K)$ , the following two conditions are equivalent:

f ∈ Lip_α(ψ, 2).
$\int_{B_{r}^{c}} {|F_{L B} f (λ, m)|}^{2} d γ_{α} (λ, m) = o (ψ (r^{- 1})) a s r \to + \infty .$

Proof. (i ⇒ ii): Let $f \in L_{α}^{2} (K)$ from (i), we have

{‖T_{(x, t)}^{(α)} f - f‖}_{2, α}^{2} = \int_{[0, + \infty [\times N} {|φ_{λ, m} (x, t) - 1|}^{2} {|F_{L B} f (λ, m)|}^{2} d γ_{α} .

Therefore, using relation (11), we have

\begin{aligned} \int_{B_{\frac{A}{| x, t |^{2}}}^{c}} {|F_{L B} f (λ, m)|}^{2} d γ_{α} & \leq C \int_{B_{\frac{A}{| x, t |^{2}}}^{c}} {|φ_{λ, m} (x, t) - 1|}^{2} {|F_{L B} f (λ, m)|}^{2} d γ_{α} \\ \leq C {‖T_{(x, t)}^{(α)} f - f‖}_{2, α}^{2} = O (ψ (| x, t |^{2})) . \end{aligned}

Consequently, (ii) holds.

(ii ⇒ i): Denote $r = \frac{η}{| x, t |^{2}}$ ⁠, by Plancherel theorem, we get

{‖T_{(x, t)}^{(α)} f - f‖}_{2, α}^{2} = I_{1} + I_{2},

where

I_{1} = \int_{B_{r}} {|φ_{λ, m} (x, t) - 1|}^{2} {|F_{L B} f (λ, m)|}^{2} d γ_{α} (λ, m)

and

I_{2} = \int_{B_{r}^{c}} {|φ_{λ, m} (x, t) - 1|}^{2} {|F_{L B} f (λ, m)|}^{2} d γ_{α} (λ, m) .

Using relation (5), we find that

I_{2} \leq 4 \int_{B_{r}^{c}} {|F_{L B} f (λ, m)|}^{2} d γ_{α} = o (ψ (r^{- 1})) = o (ψ (| x, t |^{2})) .

Denote

g (X) = \int_{X}^{\infty} {|F_{L B} f (λ, m)|}^{2} λ^{3 α + 1} d λ,

then $g^{'} (λ) = - {|F_{L B} f (λ, m)|}^{2} λ^{3 α + 1}$ ⁠. Using relation (10), which gives

I_{1} \leq C | x, t |^{4} \frac{1}{2^{2 α - 1} Γ (α + \frac{1}{2})} \sum_{m = 0}^{\infty} L_{m}^{α} (0) (\int_{0}^{\frac{η}{4 κ m | x, t |^{2}}} {(4 κ_{m})}^{2} λ^{2} (- g^{'} (λ)) d λ),

by integration by parts, we have

I_{1} \leq C | x, t |^{4} \frac{1}{2^{2 α - 1} Γ (α + \frac{1}{2})} \sum_{m = 0}^{\infty} L_{m}^{α} (0) (- \frac{η^{2}}{| x, t |^{4}} g (\frac{η}{4 κ_{m} | x, t |^{2}}) + {(4 κ_{m})}^{2} \int_{0}^{\frac{η}{4 κ_{m} | x, t |^{2}}} 2 λ g (λ) d λ) .

Remark that

\frac{1}{2^{2 α - 1} Γ (α + \frac{1}{2})} \sum_{m = 0}^{\infty} L_{m}^{α} (0) g (\frac{R}{4 κ_{m}}) = \int_{B_{R}^{c}} {|F_{L B} f (λ, m)|}^{2} d γ_{α} = o (ψ (R^{- 1})) .

Making a change of variable, one gets

\begin{aligned} I_{1} & \leq o (ψ (| x, t |^{2})) + C | x, t |^{4} \int_{0}^{\frac{η}{| x, t |^{2}}} u \frac{1}{2^{2 α - 1} Γ (α + \frac{1}{2})} \sum_{m = 0}^{\infty} L_{m}^{α} (0) g (\frac{u}{4 κ_{m}}) d u \\ = o (ψ (| x, t |^{2})) + ∣ x, t ∣^{4} o (\int_{0}^{\frac{η}{∣ x, t ∣^{2}}} u ψ (u^{- 1}) d u) \\ = o (ψ (| x, t |^{2})) . \end{aligned}

Then

{‖T_{(x, t)}^{(α)} f - f‖}_{2, α}^{2} = o (ψ (| x, t |^{2})) = o (ψ {(| x, t |)}^{2}) as | x, t | \to 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:

f is in Laguerre–Bessel–Lipschitz class Lip_α(γ, 2).
$\int_{B_{r}^{c}} {|F_{L B} f (λ, m)|}^{2} d γ_{α} (λ, m) = O (r^{- γ})$ 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.

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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

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

1. Introduction

2. Preliminaries

3. Main results

References

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

1. Introduction

2. Preliminaries

3. Main results

References

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