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`Journal of MathematicsVolume 2013 (2013), Article ID 418546, 3 pageshttp://dx.doi.org/10.1155/2013/418546`
Research Article

## Bessel Transform of -Bessel Lipschitz Functions

Department of Mathematics, Faculty of Sciences Aïn Chock, University of Hassan II, Casablanca 20100, Morocco

Received 7 January 2013; Revised 12 March 2013; Accepted 13 March 2013

Copyright © 2013 Radouan Daher and Mohamed 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.

#### Abstract

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

#### 1. Introduction and Preliminaries

Younis Theorem 5.2 [1] characterized the set of functions in 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 . Then the followings are equivalent:(1), , ,(2), where stands for the Fourier transform of .

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 ; is the Hilbert space of measurable functions on with finite norm

Let be the Bessel differential operator.

For , we introduce the Bessel normalized function of the first kind defined by where is the gamma function (see [2]).

The function satisfies the differential equation with the initial conditions and . is function infinitely differentiable, even, and, moreover, entirely analytic.

Lemma 2. For the following inequality is fulfilled: with , where is a certain constant which depends only on .

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

Lemma 3. The following inequalities are valid for Bessel function :(1), for all , (2).

Proof. See [4].

The Bessel transform we call the integral transform from [2, 5, 6]

The inverse Bessel transform is given by the formula

We have the Parseval's identity

In , consider the generalized translation operator defined by where

The following relations connect the generalized translation operator and the Bessel transform; in [7] we have

#### 2. Main Result

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

Definition 4. Let and . A function is said to be in the -Bessel Lipschitz class, denoted by Lip(, , 2), if
Our main result is as follows.

Theorem 5. Let . Then the followings are equivalents(1)  Lip. (2).

Proof. Assume that Lip(, , 2). Then we have
If then and Lemma 2 implies that
Then
We obtain where is a positive constant.
So that where since .
This proves that
Suppose now that
We write where
Estimate the summands and from above. It follows from the inequality that
To estimate , we use the inequality of Lemma 3.
Set
Using integration by parts, we obtain where 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 Sciences, 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 Zavedeniĭ. Matematika, no. 8, pp. 3–15, 2008.
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5. I. A. Kipriyanov, Singular Elliptic Boundary Value Problems, Nauka, Moscow, Russia, 1997 (Russian).
6. K. Trimèche, “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 ${L}_{2}\left({ℝ}_{+}\right)$,” Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 49, no. 7, pp. 1158–1166, 2009.