Bessel transform of (k, γ)-Bessel Lipschitz Functions

(1)

Hindawi Publishing Corporation Journal of Mathematics

Volume 2013, Article ID 418546, 3 pages http://dx.doi.org/10.1155/2013/418546

Research Article

Bessel Transform of (𝑘, 𝛾) -Bessel Lipschitz Functions

Radouan Daher and Mohamed El Hamma

Department of Mathematics, Faculty of Sciences A¨ın Chock, University of Hassan II, Casablanca 20100, Morocco

Correspondence should be addressed to Mohamed El Hamma; [email protected] Received 7 January 2013; Revised 12 March 2013; Accepted 13 March 2013

Academic Editor: Nasser Saad

Copyright © 2013 R. Daher and M. El Hamma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Using a generalized translation operator, we obtain an analog of Theorem 5.2 in Younis (1986) for the Bessel transform for functions satisfying the (k,𝛾)-Bessel Lipschitz condition in 𝐿 _2,𝛼 (R ₊ ).

1. Introduction and Preliminaries

Younis Theorem 5.2 [1] 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 transforms; namely, we have the following.

Theorem 1 (see [1]). Let 𝑓 ∈ L ² (R). Then the followings are equivalent:

(1) ‖𝑓(𝑥 + ℎ) − 𝑓(𝑥)‖ ₂ = 𝑂(ℎ ^𝛼 /(log(1/ℎ)) ^𝛽 ) 𝑎𝑠 ℎ → 0, 0 < 𝛼 < 1, 𝛽 > 0,

(2) ∫ _{|𝑥|≥𝑟} |F(𝑓)(𝑥)| ² 𝑑𝑥 = 𝑂(𝑟 ^−2𝛼 (log 𝑟) ^−2𝛽 ) 𝑎𝑠 𝑟 → +∞,

where F stands for the Fourier transform of 𝑓.

In this paper, we obtain a generalization of Theorem 1 for the Bessel transform. For this purpose, we use a generalized translation operator.

Assume that 𝐿 _2,𝛼 (R ₊ ); 𝛼 > −1/2 is the Hilbert space of measurable functions 𝑓(𝑡) on R ₊ with finite norm

󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩 2,𝛼 = (∫ ^∞

0 󵄨󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨󵄨 ² 𝑥 ^2𝛼+1 𝑑𝑥) ^1/2 . (1) Let

𝐵 = 𝑑 ²

𝑑𝑡 ² + ( 2𝛼 + 1) 𝑡

𝑑

𝑑𝑡 (2)

be the Bessel differential operator.

For 𝛼 ≥ −1/2, we introduce the Bessel normalized function of the first kind 𝑗 _𝛼 defined by

𝑗 _𝛼 (𝑧) = Γ (𝛼 + 1) ∑ ^∞

𝑛=0

(−1) ^𝑛 𝑛!Γ (𝑛 + 𝛼 + 1) ( 𝑧

2 ) ^2𝑛 , (3) where Γ is the gamma function (see [2]).

The function 𝑦 = 𝑗 _𝛼 (𝑥) satisfies the differential equation

𝐵𝑦 + 𝑦 = 0, (4)

with the initial conditions 𝑦(0) = 1 and 𝑦 ^󸀠 (0) = 0.

𝑗 _𝛼 (𝑧) is function infinitely differentiable, even, and, moreover, entirely analytic.

Lemma 2. For 𝑥 ∈ R ₊ the following inequality is fulfilled:

󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝑥)󵄨󵄨󵄨󵄨 ≥ 𝑐, (5) with 𝑥 ≥ 1, where 𝑐 > 0 is a certain constant which depends only on 𝛼.

Proof. Analog of Lemma 2.9 is in [3].

Lemma 3. The following inequalities are valid for Bessel function 𝑗 _𝛼 :

(1) |𝑗 _𝛼 (𝑥)| ≤ 1, for all 𝑥 ∈ R ⁺ , (2) 1 − 𝑗 _𝛼 (𝑥) = 𝑂(𝑥 ² ), 0 ≤ 𝑥 ≤ 1.

Proof. See [4].

(2)

2 Journal of Mathematics The Bessel transform we call the integral transform from

[2, 5, 6]

̂𝑓 (𝜆) = ∫ ^∞

0 𝑓 (𝑡) 𝑗 _𝛼 (𝜆𝑡) 𝑡 ^2𝛼+1 𝑑𝑡, 𝜆 ∈ R ⁺ . (6) The inverse Bessel transform is given by the formula

𝑓 (𝑡) = (2 ^𝛼 Γ (𝛼 + 1)) ⁻² ∫ ^∞

0 ̂𝑓 (𝜆) 𝑗 _𝛼 (𝜆𝑡) 𝜆 ^2𝛼+1 𝑑𝜆. (7) We have the Parseval’s identity

󵄩󵄩󵄩󵄩 󵄩̂𝑓󵄩󵄩󵄩󵄩󵄩 2,𝛼 = 2 ^𝛼 Γ (𝛼 + 1) 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩 2,𝛼 . (8) In 𝐿 _2,𝛼 (R ₊ ), consider the generalized translation operator 𝑇 _ℎ defined by

𝑇 _ℎ 𝑓 (𝑡) = 𝑐 _𝛼 ∫ ^𝜋

0 𝑓 (√𝑡 ² + ℎ ² − 2𝑡ℎ cos 𝜑) sin ^2𝛼 𝜑 𝑑𝜑, (9) where

𝑐 _𝛼 = (∫ ^𝜋

0 sin ^2𝛼 𝜑 𝑑𝜑) ⁻¹ = Γ (𝛼 + 1)

Γ (1/2) Γ (𝛼 + (1/2)) . (10) The following relations connect the generalized transla- tion operator and the Bessel transform; in [7] we have

(̂ T _ℎ 𝑓) (𝜆) = 𝑗 _𝛼 (𝜆ℎ) ̂𝑓 (𝜆) . (11)

2. Main Result

In this section we give the main result of this paper. We need first to define (𝑘, 𝛾)-Bessel Lipschitz class.

Definition 4. Let 0 < 𝑘 < 1 and 𝛾 ≥ 0. A function 𝑓 ∈ 𝐿 _2,𝛼 (R ⁺ ) is said to be in the (𝑘, 𝛾)-Bessel Lipschitz class, denoted by Lip(𝑘, 𝛾, 2), if

󵄩󵄩󵄩󵄩 T _ℎ 𝑓(𝑡) − 𝑓(𝑡)󵄩󵄩󵄩󵄩 2,𝛼 = 𝑂 ( ℎ ^𝑘

(log (1/ℎ)) ^𝛾 ) , as ℎ 󳨀→ 0.

(12) Our main result is as follows.

Theorem 5. Let 𝑓 ∈ 𝐿 _2,𝛼 (R ⁺ ). Then the followings are equivalents

(1) 𝑓 ∈ Lip(𝑘, 𝛾, 2).

(2) ∫ _𝑟 ^∞ |̂𝑓(𝜆)| ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝑂(𝑟 ^−2𝑘 /(log 𝑟) ^2𝛾 ), 𝑎𝑠 𝑟 → +∞.

Proof. (1) ⇒ (2) Assume that 𝑓 ∈ Lip(𝑘, 𝛾, 2). Then we have

󵄩󵄩󵄩󵄩 T _ℎ 𝑓(𝑡) − 𝑓(𝑡)󵄩󵄩󵄩󵄩 ² 2,𝛼

= 1

(2 ^𝛼 Γ (𝛼 + 1)) ² ∫ ^∞

0 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆.

(13)

If 𝜆 ∈ [1/ℎ, 2/ℎ] then 𝜆ℎ ≥ 1 and Lemma 2 implies that 1 ≤ 1

𝑐 ² 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 . (14) Then

∫ ^2/ℎ

1/ℎ 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

≤ 1 𝑐 ² ∫ ^2/ℎ

1/ℎ 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

≤ 1 𝑐 ² ∫ ^∞

0 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

= 𝑂 ( ℎ ^2𝑘 (log (1/ℎ)) ^2𝛾 ) .

(15)

We obtain

∫ ^2𝑟

𝑟 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 ≤ 𝐶 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 , (16) where 𝐶 is a positive constant.

So that

∫ ^∞

𝑟 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

= [∫ ^2𝑟

𝑟 + ∫ ^4𝑟

2𝑟 + ∫ ^8𝑟

4𝑟 + ⋅ ⋅ ⋅] 󵄨󵄨󵄨󵄨󵄨̂𝑓(𝜆) 󵄨󵄨󵄨󵄨 󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

≤ 𝐶 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 + 𝐶 ( 2𝑟) ^−2𝑘

(log 2𝑟) ^2𝛾 + 𝐶 ( 4𝑟) ^−2𝑘 (log 4𝑟) ^2𝛾 + ⋅ ⋅ ⋅

≤ 𝐶 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 (1 + 2 ^−2𝑘 + (2 ^−2𝑘 ) ² + (2 ^−2𝑘 ) ³ + ⋅ ⋅ ⋅)

≤ 𝐶𝐾 𝑟 ^−2𝑘 (log 𝑟) ^2𝛾 ,

(17) where 𝐾 = (1 − 2 ^−2𝑘 ) ⁻¹ since 2 ^−2𝑘 < 1.

This proves that

∫ ^∞

𝑟 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝑂 ( 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 ) as 𝑟 󳨀→ +∞.

(18) (2) ⇒ (1) Suppose now that

∫ ^∞

𝑟 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝑂 ( 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 ) as 𝑟 󳨀→ +∞.

(19) We write

∫ ^∞

0 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝐼 ₁ + 𝐼 ₂ , (20)

(3)

Journal of Mathematics 3 where

𝐼 ₁ = ∫ ^1/ℎ

0 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆, 𝐼 ₂ = ∫ ^∞

1/ℎ 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆.

(21)

Estimate the summands 𝐼 ₁ and 𝐼 ₂ from above. It follows from the inequality |𝑗 _𝛼 (𝜆ℎ)| ≤ 1 that

𝐼 ₂ = ∫ ^∞

1/ℎ 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

≤ 4 ∫ ^∞

1/ℎ 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝑂 ( ℎ ^2𝑘 (log (1/ℎ)) ^2𝛾 ) .

(22)

To estimate 𝐼 ₁ , we use the inequality (2) of Lemma 3.

Set

𝜙 (𝑥) = ∫ ^∞

𝑥 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆. (23) Using integration by parts, we obtain

𝐼 ₁ ≤ −𝐶 ₁ ℎ ² ∫ ^1/ℎ

0 𝑠 ² 𝜙 ^󸀠 (𝑠) 𝑑𝑠

≤ −𝐶 ₁ 𝜙 ( 1

ℎ ) + 2𝐶 ₁ ℎ ² ∫ ^1/ℎ

0 𝑠𝜙 (𝑠) 𝑑𝑠

≤ 𝐶 ₂ ℎ ² ∫ ^1/ℎ

0 𝑠𝜙 (𝑠) 𝑑𝑠

≤ 𝐶 ₂ ℎ ² ∫ ^1/ℎ

0 𝑠𝑠 ^−2𝑘 (log 𝑠) ^−2𝛾 𝑑𝑠

≤ 𝐶 ₃ ℎ ^2𝑘 (log (1/ℎ)) ^−2𝛾 ,

(24)

where 𝐶 ₁ , 𝐶 ₂ , and 𝐶 ₂ are positive constants and this ends the proof.

References

[1] M. S. Younis, “Fourier transforms of Dini-Lipschitz functions,”

International Journal of Mathematics and Mathematical Sci- ences, vol. 9, no. 2, pp. 301–312, 1986.

[2] B. M. Levitan, “Expansion in Fourier series and integrals with Bessel functions,” Uspekhi Matematicheskikh Nauk, vol. 6, no. 2, pp. 102–143, 1951 (Russian).

[3] E. S. Belkina and S. S. Platonov, “Equivalence of K-functionals and moduli of smoothness constructed by generalized Dunkl translations,” Izvestiya Vysshikh Uchebnykh Zavedeni˘ı. Matem- atika, no. 8, pp. 3–15, 2008.

[4] V. A. Abilov and F. V. Abilova, “Approximation of functions by Fourier-Bessel sums,” Izvestiya Vysshikh Uchebnykh Zavedeni˘ı.

Matematika, no. 8, pp. 3–9, 2001 (Russian).

[5] I. A. Kipriyanov, Singular Elliptic Boundary Value Problems, Nauka, Moscow, Russia, 1997 (Russian).

[6] K. Trim`eche, “Transmutation operators and mean-periodic functions associated with differential operators,” Mathematical Reports, vol. 4, no. 1, pp. 1–282, 1988.

[7] V. A. Abilov, F. V. Abilova, and M. K. Kerimov, “On some

estimates for the Fourier-Bessel integral transform in the space

𝐿 ₂ (R ₊ ),” Zhurnal Vychislitel’noi Matematiki i Matematicheskoi

Fiziki, vol. 49, no. 7, pp. 1158–1166, 2009.

(4)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics ^{Journal of}

Mathematical Problems in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics ^{Journal of}

Probability and Statistics

Journal of

Mathematical Physics Advances in

Complex Analysis ^{Journal of}

Optimization ^{Journal of}

Combinatorics

International Journal of

Operations Research Advances in

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Algebra

Discrete Dynamics in Nature and Society

Decision Sciences Advances in

Discrete Mathematics ^{Journal of}

http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation

Stochastic Analysis

International Journal of

Bessel transform of (k, γ)-Bessel Lipschitz Functions

Hindawi Publishing Corporation Journal of Mathematics

Volume 2013, Article ID 418546, 3 pages http://dx.doi.org/10.1155/2013/418546

Research Article

Bessel Transform of (𝑘, 𝛾) -Bessel Lipschitz Functions

Radouan Daher and Mohamed El Hamma

Department of Mathematics, Faculty of Sciences A¨ın Chock, University of Hassan II, Casablanca 20100, Morocco

Correspondence should be addressed to Mohamed El Hamma; [email protected] Received 7 January 2013; Revised 12 March 2013; Accepted 13 March 2013

Academic Editor: Nasser Saad

Copyright © 2013 R. Daher and M. El Hamma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Using a generalized translation operator, we obtain an analog of Theorem 5.2 in Younis (1986) for the Bessel transform for functions satisfying the (k,𝛾)-Bessel Lipschitz condition in 𝐿 2,𝛼 (R + ).

1. Introduction and Preliminaries

Younis Theorem 5.2 [1] 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 transforms; namely, we have the following.

Theorem 1 (see [1]). Let 𝑓 ∈ L 2 (R). Then the followings are equivalent:

(1) ‖𝑓(𝑥 + ℎ) − 𝑓(𝑥)‖ 2 = 𝑂(ℎ 𝛼 /(log(1/ℎ)) 𝛽 ) 𝑎𝑠 ℎ → 0, 0 < 𝛼 < 1, 𝛽 > 0,

(2) ∫ |𝑥|≥𝑟 |F(𝑓)(𝑥)| 2 𝑑𝑥 = 𝑂(𝑟 −2𝛼 (log 𝑟) −2𝛽 ) 𝑎𝑠 𝑟 → +∞,

where F stands for the Fourier transform of 𝑓.

In this paper, we obtain a generalization of Theorem 1 for the Bessel transform. For this purpose, we use a generalized translation operator.

Assume that 𝐿 2,𝛼 (R + ); 𝛼 > −1/2 is the Hilbert space of measurable functions 𝑓(𝑡) on R + with finite norm

󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩 2,𝛼 = (∫ ∞

0 󵄨󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨󵄨 2 𝑥 2𝛼+1 𝑑𝑥) 1/2 . (1) Let

𝐵 = 𝑑 2

𝑑𝑡 2 + ( 2𝛼 + 1) 𝑡

𝑑

𝑑𝑡 (2)

be the Bessel differential operator.

For 𝛼 ≥ −1/2, we introduce the Bessel normalized function of the first kind 𝑗 𝛼 defined by

𝑗 𝛼 (𝑧) = Γ (𝛼 + 1) ∑ ∞

𝑛=0

(−1) 𝑛 𝑛!Γ (𝑛 + 𝛼 + 1) ( 𝑧

2 ) 2𝑛 , (3) where Γ is the gamma function (see [2]).

The function 𝑦 = 𝑗 𝛼 (𝑥) satisfies the differential equation

𝐵𝑦 + 𝑦 = 0, (4)

with the initial conditions 𝑦(0) = 1 and 𝑦 󸀠 (0) = 0.

𝑗 𝛼 (𝑧) is function infinitely differentiable, even, and, moreover, entirely analytic.

Lemma 2. For 𝑥 ∈ R + the following inequality is fulfilled:

󵄨󵄨󵄨󵄨1 − 𝑗 𝛼 (𝑥)󵄨󵄨󵄨󵄨 ≥ 𝑐, (5) with 𝑥 ≥ 1, where 𝑐 > 0 is a certain constant which depends only on 𝛼.

Proof. Analog of Lemma 2.9 is in [3].

Lemma 3. The following inequalities are valid for Bessel function 𝑗 𝛼 :

(1) |𝑗 𝛼 (𝑥)| ≤ 1, for all 𝑥 ∈ R + , (2) 1 − 𝑗 𝛼 (𝑥) = 𝑂(𝑥 2 ), 0 ≤ 𝑥 ≤ 1.

Proof. See [4].

2 Journal of Mathematics The Bessel transform we call the integral transform from

[2, 5, 6]

̂𝑓 (𝜆) = ∫ ∞

0 𝑓 (𝑡) 𝑗 𝛼 (𝜆𝑡) 𝑡 2𝛼+1 𝑑𝑡, 𝜆 ∈ R + . (6) The inverse Bessel transform is given by the formula

𝑓 (𝑡) = (2 𝛼 Γ (𝛼 + 1)) −2 ∫ ∞

0 ̂𝑓 (𝜆) 𝑗 𝛼 (𝜆𝑡) 𝜆 2𝛼+1 𝑑𝜆. (7) We have the Parseval’s identity

󵄩󵄩󵄩󵄩 󵄩̂𝑓󵄩󵄩󵄩󵄩󵄩 2,𝛼 = 2 𝛼 Γ (𝛼 + 1) 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩 2,𝛼 . (8) In 𝐿 2,𝛼 (R + ), consider the generalized translation operator 𝑇 ℎ defined by

𝑇 ℎ 𝑓 (𝑡) = 𝑐 𝛼 ∫ 𝜋

0 𝑓 (√𝑡 2 + ℎ 2 − 2𝑡ℎ cos 𝜑) sin 2𝛼 𝜑 𝑑𝜑, (9) where

𝑐 𝛼 = (∫ 𝜋

0 sin 2𝛼 𝜑 𝑑𝜑) −1 = Γ (𝛼 + 1)

Γ (1/2) Γ (𝛼 + (1/2)) . (10) The following relations connect the generalized transla- tion operator and the Bessel transform; in [7] we have

(̂ T ℎ 𝑓) (𝜆) = 𝑗 𝛼 (𝜆ℎ) ̂𝑓 (𝜆) . (11)

2. Main Result

In this section we give the main result of this paper. We need first to define (𝑘, 𝛾)-Bessel Lipschitz class.

Definition 4. Let 0 < 𝑘 < 1 and 𝛾 ≥ 0. A function 𝑓 ∈ 𝐿 2,𝛼 (R + ) is said to be in the (𝑘, 𝛾)-Bessel Lipschitz class, denoted by Lip(𝑘, 𝛾, 2), if

󵄩󵄩󵄩󵄩 T ℎ 𝑓(𝑡) − 𝑓(𝑡)󵄩󵄩󵄩󵄩 2,𝛼 = 𝑂 ( ℎ 𝑘

(log (1/ℎ)) 𝛾 ) , as ℎ 󳨀→ 0.

(12) Our main result is as follows.

Theorem 5. Let 𝑓 ∈ 𝐿 2,𝛼 (R + ). Then the followings are equivalents

(1) 𝑓 ∈ Lip(𝑘, 𝛾, 2).

(2) ∫ 𝑟 ∞ |̂𝑓(𝜆)| 2 𝜆 2𝛼+1 𝑑𝜆 = 𝑂(𝑟 −2𝑘 /(log 𝑟) 2𝛾 ), 𝑎𝑠 𝑟 → +∞.

Proof. (1) ⇒ (2) Assume that 𝑓 ∈ Lip(𝑘, 𝛾, 2). Then we have

󵄩󵄩󵄩󵄩 T ℎ 𝑓(𝑡) − 𝑓(𝑡)󵄩󵄩󵄩󵄩 2 2,𝛼

= 1

(2 𝛼 Γ (𝛼 + 1)) 2 ∫ ∞

0 󵄨󵄨󵄨󵄨1 − 𝑗 𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 2 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 2 𝜆 2𝛼+1 𝑑𝜆.

(13)

If 𝜆 ∈ [1/ℎ, 2/ℎ] then 𝜆ℎ ≥ 1 and Lemma 2 implies that 1 ≤ 1

𝑐 2 󵄨󵄨󵄨󵄨1 − 𝑗 𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 . (14) Then

∫ 2/ℎ

1/ℎ 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 2 𝜆 2𝛼+1 𝑑𝜆

≤ 1 𝑐 2 ∫ 2/ℎ

1/ℎ 󵄨󵄨󵄨󵄨1 − 𝑗 𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 2 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 2 𝜆 2𝛼+1 𝑑𝜆

≤ 1 𝑐 2 ∫ ∞

0 󵄨󵄨󵄨󵄨1 − 𝑗 𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 2 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 2 𝜆 2𝛼+1 𝑑𝜆

= 𝑂 ( ℎ 2𝑘 (log (1/ℎ)) 2𝛾 ) .

(15)

We obtain

Using a generalized translation operator, we obtain an analog of Theorem 5.2 in Younis (1986) for the Bessel transform for functions satisfying the (k,𝛾)-Bessel Lipschitz condition in 𝐿 _2,𝛼 (R ₊ ).

Younis Theorem 5.2 [1] 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 transforms; namely, we have the following.

Theorem 1 (see [1]). Let 𝑓 ∈ L ² (R). Then the followings are equivalent:

(1) ‖𝑓(𝑥 + ℎ) − 𝑓(𝑥)‖ ₂ = 𝑂(ℎ ^𝛼 /(log(1/ℎ)) ^𝛽 ) 𝑎𝑠 ℎ → 0, 0 < 𝛼 < 1, 𝛽 > 0,

(2) ∫ _{|𝑥|≥𝑟} |F(𝑓)(𝑥)| ² 𝑑𝑥 = 𝑂(𝑟 ^−2𝛼 (log 𝑟) ^−2𝛽 ) 𝑎𝑠 𝑟 → +∞,

Assume that 𝐿 _2,𝛼 (R ₊ ); 𝛼 > −1/2 is the Hilbert space of measurable functions 𝑓(𝑡) on R ₊ with finite norm

󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩 2,𝛼 = (∫ ^∞

0 󵄨󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨󵄨 ² 𝑥 ^2𝛼+1 𝑑𝑥) ^1/2 . (1) Let

𝐵 = 𝑑 ²

𝑑𝑡 ² + ( 2𝛼 + 1) 𝑡

For 𝛼 ≥ −1/2, we introduce the Bessel normalized function of the first kind 𝑗 _𝛼 defined by

𝑗 _𝛼 (𝑧) = Γ (𝛼 + 1) ∑ ^∞

(−1) ^𝑛 𝑛!Γ (𝑛 + 𝛼 + 1) ( 𝑧

2 ) ^2𝑛 , (3) where Γ is the gamma function (see [2]).

The function 𝑦 = 𝑗 _𝛼 (𝑥) satisfies the differential equation

with the initial conditions 𝑦(0) = 1 and 𝑦 ^󸀠 (0) = 0.

𝑗 _𝛼 (𝑧) is function infinitely differentiable, even, and, moreover, entirely analytic.

Lemma 2. For 𝑥 ∈ R ₊ the following inequality is fulfilled:

󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝑥)󵄨󵄨󵄨󵄨 ≥ 𝑐, (5) with 𝑥 ≥ 1, where 𝑐 > 0 is a certain constant which depends only on 𝛼.

Lemma 3. The following inequalities are valid for Bessel function 𝑗 _𝛼 :

(1) |𝑗 _𝛼 (𝑥)| ≤ 1, for all 𝑥 ∈ R ⁺ , (2) 1 − 𝑗 _𝛼 (𝑥) = 𝑂(𝑥 ² ), 0 ≤ 𝑥 ≤ 1.

̂𝑓 (𝜆) = ∫ ^∞

0 𝑓 (𝑡) 𝑗 _𝛼 (𝜆𝑡) 𝑡 ^2𝛼+1 𝑑𝑡, 𝜆 ∈ R ⁺ . (6) The inverse Bessel transform is given by the formula

𝑓 (𝑡) = (2 ^𝛼 Γ (𝛼 + 1)) ⁻² ∫ ^∞

0 ̂𝑓 (𝜆) 𝑗 _𝛼 (𝜆𝑡) 𝜆 ^2𝛼+1 𝑑𝜆. (7) We have the Parseval’s identity

󵄩󵄩󵄩󵄩 󵄩̂𝑓󵄩󵄩󵄩󵄩󵄩 2,𝛼 = 2 ^𝛼 Γ (𝛼 + 1) 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩 2,𝛼 . (8) In 𝐿 _2,𝛼 (R ₊ ), consider the generalized translation operator 𝑇 _ℎ defined by

𝑇 _ℎ 𝑓 (𝑡) = 𝑐 _𝛼 ∫ ^𝜋

0 𝑓 (√𝑡 ² + ℎ ² − 2𝑡ℎ cos 𝜑) sin ^2𝛼 𝜑 𝑑𝜑, (9) where

𝑐 _𝛼 = (∫ ^𝜋

0 sin ^2𝛼 𝜑 𝑑𝜑) ⁻¹ = Γ (𝛼 + 1)

(̂ T _ℎ 𝑓) (𝜆) = 𝑗 _𝛼 (𝜆ℎ) ̂𝑓 (𝜆) . (11)

Definition 4. Let 0 < 𝑘 < 1 and 𝛾 ≥ 0. A function 𝑓 ∈ 𝐿 _2,𝛼 (R ⁺ ) is said to be in the (𝑘, 𝛾)-Bessel Lipschitz class, denoted by Lip(𝑘, 𝛾, 2), if

󵄩󵄩󵄩󵄩 T _ℎ 𝑓(𝑡) − 𝑓(𝑡)󵄩󵄩󵄩󵄩 2,𝛼 = 𝑂 ( ℎ ^𝑘

(log (1/ℎ)) ^𝛾 ) , as ℎ 󳨀→ 0.

Theorem 5. Let 𝑓 ∈ 𝐿 _2,𝛼 (R ⁺ ). Then the followings are equivalents

(2) ∫ _𝑟 ^∞ |̂𝑓(𝜆)| ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝑂(𝑟 ^−2𝑘 /(log 𝑟) ^2𝛾 ), 𝑎𝑠 𝑟 → +∞.

󵄩󵄩󵄩󵄩 T _ℎ 𝑓(𝑡) − 𝑓(𝑡)󵄩󵄩󵄩󵄩 ² 2,𝛼

(2 ^𝛼 Γ (𝛼 + 1)) ² ∫ ^∞

0 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆.

𝑐 ² 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 . (14) Then

∫ ^2/ℎ

1/ℎ 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

≤ 1 𝑐 ² ∫ ^2/ℎ

1/ℎ 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

≤ 1 𝑐 ² ∫ ^∞

0 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

= 𝑂 ( ℎ ^2𝑘 (log (1/ℎ)) ^2𝛾 ) .

∫ ^2𝑟

𝑟 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 ≤ 𝐶 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 , (16) where 𝐶 is a positive constant.

∫ ^∞

𝑟 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

= [∫ ^2𝑟

𝑟 + ∫ ^4𝑟

2𝑟 + ∫ ^8𝑟

4𝑟 + ⋅ ⋅ ⋅] 󵄨󵄨󵄨󵄨󵄨̂𝑓(𝜆) 󵄨󵄨󵄨󵄨 󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

≤ 𝐶 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 + 𝐶 ( 2𝑟) ^−2𝑘

(log 2𝑟) ^2𝛾 + 𝐶 ( 4𝑟) ^−2𝑘 (log 4𝑟) ^2𝛾 + ⋅ ⋅ ⋅

≤ 𝐶 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 (1 + 2 ^−2𝑘 + (2 ^−2𝑘 ) ² + (2 ^−2𝑘 ) ³ + ⋅ ⋅ ⋅)

≤ 𝐶𝐾 𝑟 ^−2𝑘 (log 𝑟) ^2𝛾 ,

(17) where 𝐾 = (1 − 2 ^−2𝑘 ) ⁻¹ since 2 ^−2𝑘 < 1.

∫ ^∞

𝑟 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝑂 ( 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 ) as 𝑟 󳨀→ +∞.

∫ ^∞

𝑟 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝑂 ( 𝑟 ^−2𝑘

(log 𝑟) ^2𝛾 ) as 𝑟 󳨀→ +∞.

∫ ^∞

0 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝐼 ₁ + 𝐼 ₂ , (20)

𝐼 ₁ = ∫ ^1/ℎ

0 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨 ̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆, 𝐼 ₂ = ∫ ^∞

1/ℎ 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆.

Estimate the summands 𝐼 ₁ and 𝐼 ₂ from above. It follows from the inequality |𝑗 _𝛼 (𝜆ℎ)| ≤ 1 that

𝐼 ₂ = ∫ ^∞

1/ℎ 󵄨󵄨󵄨󵄨1 − 𝑗 ^𝛼 (𝜆ℎ)󵄨󵄨󵄨󵄨 ² 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆

≤ 4 ∫ ^∞

1/ℎ 󵄨󵄨󵄨󵄨 󵄨̂𝑓 (𝜆)󵄨󵄨󵄨󵄨󵄨 ² 𝜆 ^2𝛼+1 𝑑𝜆 = 𝑂 ( ℎ ^2𝑘 (log (1/ℎ)) ^2𝛾 ) .

To estimate 𝐼 ₁ , we use the inequality (2) of Lemma 3.