Journal of Mathematics

Volume 2013, Article ID 418546, 3 pages

http://dx.doi.org/10.1155/2013/418546

## 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

Academic Editor: Nasser Saad

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

- M. S. Younis, “Fourier transforms of Dini-Lipschitz functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 9, no. 2, pp. 301–312, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. M. Levitan, “Expansion in Fourier series and integrals with Bessel functions,”
*Uspekhi Matematicheskikh Nauk*, vol. 6, no. 2, pp. 102–143, 1951 (Russian). View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - V. A. Abilov and F. V. Abilova, “Approximation of functions by Fourier-Bessel sums,”
*Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika*, no. 8, pp. 3–9, 2001 (Russian). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. A. Kipriyanov,
*Singular Elliptic Boundary Value Problems*, Nauka, Moscow, Russia, 1997 (Russian). View at MathSciNet - K. Trimèche, “Transmutation operators and mean-periodic functions associated with differential operators,”
*Mathematical Reports*, vol. 4, no. 1, pp. 1–282, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. A. Abilov, F. V. Abilova, and M. K. Kerimov, “On some estimates for the Fourier-Bessel integral transform in the space ${L}_{2}({\mathbb{R}}_{+})$,”
*Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki*, vol. 49, no. 7, pp. 1158–1166, 2009. View at Publisher · View at Google Scholar · View at MathSciNet