International Journal of Mathematics and Mathematical Sciences

1. Introduction

The uncertainty principle is a metatheorem in harmonic analysis that asserts, with the use of some inequalities, that a function and its Fourier transform cannot be sharply localized. We refer to the survey article by Folland and Sitaram [1] and the book of Havin and Jöricke [2] for various classical uncertainty principles of different nature which may be found in the literature.

In [3], the authors gave -analogues of the Heisenberg uncertainty principle for the -Fourier-cosine and the -Fourier-sine transforms. One of the aims of this paper is to provide a generalization of their work next to state local uncertainty principles for various -Fourier transforms.

This paper is organized as follows. In Section 2, we present some preliminaries results and notations that will be useful in the sequel. In Section 3, we prove a density theorem and a -analogue of the Hausdorff-Young inequality. Then, we state a generalization of the -Heisenberg uncertainty principle for the -Fourier-cosine and the -Fourier-sine transforms. In Section 4, we state local uncertainty principles for the -Fourier-cosine, -Fourier-sine, and -Bessel-Fourier transforms. Then, we give a Heisenberg-Weyl-type inequality for some -Bessel-Fourier transform.

2. Notations and Preliminaries

Throughout this paper, we assume . We recall some usual notions and notations used in the -theory (see [4, 5]). We refer to the book by Gasper and Rahman [4] for the definitions, notations, and properties of the -shifted factorials and the -hypergeometric functions.

We write , , andThe -derivative of a function is given by, provided that the limit exists.

The -Jackson integrals from to and from to , of a function , are (see [6])provided that the sums converge absolutely.

The -Jackson integral in a generic interval is given by (see [6])The -integration by parts rule is given, for suitable functions and by

Jackson (see [6]) defined a -analogue of the Gamma function by

The third Jackson (2.6)-Bessel function (see [,]) is and the -trigonometric functions (-cosine and -sine) are defined by (see [9])They verify We need the following spaces and norms.

(i) is the space of even functions on satisfying (ii), , , is the set of all functions defined on such that(iii), , and (iv) is the set of all bounded functions on . We write

3. Generalization of the Heisenberg Uncertainty Principle

The -Fourier-cosine and the -Fourier-sine transforms are defined as (see [8, 9])whereLetting subject to the condition gives, at least formally, the classical Fourier transforms (see [3, 10]). In the remainder of the present paper, we assume that this condition holds.

It was shown in [8, 9] that we have the following result.

Proposition 3.1. (1) For , one has and
(2) is an isomorphism of (resp., ) onto itself. Moreover, one has and the following Plancherel formula:

Similarly, it was shown in [3, 8] that the -Fourier-sine transform verifies the following properties.

Proposition 3.2. (1) For , one has and
(2) is an isomorphism of onto itself; its inverse is given by . One has the following Plancherel formula:

Let us now state the following useful density result.

Proposition 3.3. For all , is dense in .

Proof. Let and . For , put , where is the characteristic function of .
It is clear that for all , and . So, the Lebesgue theorem implies that converges to in .

Remark 3.4. Using the density of in (), one can see that the -Fourier-cosine (resp., -Fourier-sine) transform has a unique continuous extension on , that will also be denoted as (resp., ). We have the following -analogue of the Hausdorff-Young inequality.

Theorem 3.5. Let (resp., ) and (resp., ) be the dual exponent of . For all in , the functions and belong to , and one has where

Proof. The result is a direct consequence of [11, Theorem 1.3.4, page 35], and Propositions 3.1 and 3.2, by taking as a set of simple functions.

The following lemma gives relations between the two Fourier -trigonometric transforms.

Lemma 3.6. (1) For such that , one has
(2) Additionally, if , then

Proof. The same steps as in the proof of [3, Lemma 2]; the -integration by parts rule and the fact thatgive the result.

In [3], the authors proved the following -analogues of the Heisenberg uncertainty principle.

Theorem 3.7. Let be in such that is in . Then, In addition, if , one has

Now, we are in a position to generalize Theorem 3.7. One obvious way to generalize it is to replace the norms by norms. This is the purpose of the following result.

Theorem 3.8. For and , one has where with and being given by (3.8).

Proof. The case has been dealt with in Theorem 3.7. Now, assume and let be the dual exponent of . Let such that . From the relationthe -integration by parts rule, and the Hölder inequality, we have, since tends to as tends to in ,However, the change of variable givesSo,On the other hand, we have since is in . Then, by using Lemma 3.6 and the -analogue of the Hausdorff-Young inequality, we obtain Thus,Now, let ; it is easy to see that for all , , , and converges to in . Moreover, if the right-hand side of (3.14) (resp., (3.15)) is finite, then the functions and (resp., ) are in and they are limits in (as tends to ) of and (resp., ), respectively. Finally, the substitution of in (3.22) and a passage to the limit when tends to complete the proof.

4. Local Uncertainty Principles

In the literature, the first classical local inequalities were obtained by Faris (see [12]) in 1978, and they were generalized by Price (see [13, 14]) in 1983 and 1987. In this section, we will generalize Price's results by giving their -analogues.

4.1. Local Uncertainty Principles for the -Fourier Trigonometric Transforms

Theorem 4.1. If , there is a constant such that for all bounded subset of and all , one has Here, and , where Proof. For , let be the characteristic function of and .
Then, for , we have, since ,and by the use of the Hölder inequality, we obtainOn the other hand, since ), we have , and by the Plancherel formula, we getSo,The desired result is obtained by minimizing the right-hand side of the previous inequality over .

Corollary 4.2. For and , there is a constant such that for all , one has

Proof. For , put and . It is easy to see that is a bounded subset of and
Then, from the Plancherel formula and Theorem 4.1, we haveChoosing so as to minimize the right-hand side of the inequality, we obtain , with , and is the constant given in Theorem 4.1.

In the same way, one can prove the following local uncertainty principle for the -Fourier-sine transform.

Theorem 4.3. If , there is a constant such that for all bounded subset of and all , one has where .

Corollary 4.4. For and , there is a constant such that for all , one has with

Proof. The same steps of Corollary 4.2 give the result.

Theorem 4.5. If , there is a constant such that for all bounded subset of and , one has

The proof of this result needs the following lemmas.

Lemma 4.6. Suppose , then for all such that , where .

Proof. From [15, Example 1], and the Hölder inequality, we havewhere .

Lemma 4.7. Suppose , then for all , such that , one has where

Proof. For define the function by .
We have ,
Replacement of by in Lemma 4.6 givesNow, for all put We have and Then, for all The right-hand side of this inequality is minimized by choosingWhen this is done, we obtain the result.

Proof of Theorem 4.5. Since the proofs of the two statements are similar, it is sufficient to prove (4.11).
Let be a bounded subset of . When the right-hand side of the inequality (4.11) is finite, Lemma 4.6 implies that ; so is defined and bounded on . Using Proposition 3.1, Lemma 4.7, and the fact thatwe obtain the result with .