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.