International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 465909, 13 pages
http://dx.doi.org/10.1155/2008/465909
On Some Inequalities of Uncertainty Principles Type in Quantum Calculus
1Faculté des Sciences de Tunis, Université de Tunis El Manar, 1060 Tunis, Tunisia
2Institut Préparatoire aux Études d'Ingénieur de Monastir, Université de Monastir, 5000 Monastir, Tunisia
3Institut de Biotechnologie, Université de Jendouba, 9000 Béja, Tunisia
Received 19 July 2007; Revised 31 January 2008; Accepted 14 April 2008
Academic Editor: Wolfgang Castell
Copyright © 2008 Ahmed Fitouhi et al. 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
The aim of this paper is to generalize the -Heisenberg uncertainty principles studied by Bettaibi et al. (2007), to state local uncertainty principles for the -Fourier-cosine, the -Fourier-sine, and the -Bessel-Fourier transforms, then to provide an inequality of Heisenberg-Weyl-type for the -Bessel-Fourier transform.
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 .
Remark 4.8. By the same technique as in the proof of Corollary 4.2, we can show that Theorem 4.5 leads to inequalities (4.6) and (4.9) with some different constants.
4.2. Local Uncertainty Principles for the -Bessel-Fourier Transform
The -Bessel-Fourier transform is defined (see [16]) for bywhereis the normalized third Jackson -Bessel function, andIt was shown in [10] that for , we have the following result.
Theorem 4.9. (1) For ,
one has and
(2) is an isomorphism of onto itself, ,
and one has the following Plancherel formula:
The following
result states a local uncertainty principle for the -Bessel-Fourier transform.
Theorem 4.10. For and , there is a constant such that for all and all bounded subset of , one has Here, , , and
ProofLet (4.25), (4.25), (4.25),
and let (4.25) be a bounded subset of (4.25).
For (4.25),
we have, since (4.25),
Proof. Let , , ,
and let be a bounded subset of .
For ,
we have, since ,However, by the use of the
Hölder inequality, we obtainNow, if is the integer such that ,
we get, since ,Then,On the other hand, since ,
we have and by the Plancherel formula (4.23), we
obtainSo,By minimization of the
right-hand side of the previous inequality over and by easy computation, we obtain the desired
result.
Theorem 4.11. For and , there exists a constant such that for all bounded subset of and all in , one has
Proof. Since , the same steps as in the proof of Theorem 4.5 and the relation (4.22) give the result with
Corollary 4.12. For and , there is a constant such that for all , one has with where (resp., ) is the constant given in Theorem 4.10 (resp., Theorem 4.11).
Proof. For ,
we put and .
We have is a bounded subset of and Then, the Plancherel formula (4.23) and Theorems
4.10 and 4.11 lead to The desired result follows by
minimizing the right expressions over .
Remark that when , we obtain a Heisenberg-Weyl-type inequality for the -Bessel-Fourier transform.
Corollary 4.13. For , one has for all ,
Acknowledgment
The authors would like to thank the reviewers for their helpful remarks and constructive criticism.
References
- G. B. Folland and A. Sitaram, “The uncertainty principle: a mathematical survey,” Journal of Fourier Analysis and Applications, vol. 3, no. 3, pp. 207–238, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- V. Havin and B. Jöricke, The Uncertainty Principle in Harmonic Analysis, vol. 28 of Results in Mathematics and Related Areas (3), Springer, Berlin, Germany, 1994. View at Zentralblatt MATH · View at MathSciNet
- N. Bettaibi, A. Fitouhi, and W. Binous, “Uncertainty principles for the -trigonometric Fourier transforms,” Mathematical Sciences Research Journal, vol. 11, no. 7, pp. 469–479, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 96 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2nd edition, 2004. View at Zentralblatt MATH · View at MathSciNet
- V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002. View at Zentralblatt MATH · View at MathSciNet
- F. H. Jackson, “On a -definite integrals,” Quarterly Journal of Pure and Applied Mathematics, vol. 41, pp. 193–203, 1910. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- M. E. H. Ismail, “On Jacksons third -Bessel function and -exponentials,” preprint, 2001. View at Zentralblatt MATH · View at MathSciNet
- T. H. Koornwinder and R. F. Swarttouw, “On -analogues of the Fourier and Hankel transforms,” Transactions of the American Mathematical Society, vol. 333, no. 1, pp. 445–461, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- F. Bouzeffour, q-Analyse harmonique, Ph.D. thesis, Faculty of Sciences of Tunis, Tunis, Tunisia, 2002. View at Zentralblatt MATH · View at MathSciNet
- A. Fitouhi, N. Bettaibi, and W. Binous, “Inversion formulas for the -Riemann-Liouville and -Weyl transforms using wavelets,” Fractional Calculus & Applied Analysis, vol. 10, no. 4, pp. 327–342, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, NJ, USA, 2004. View at Zentralblatt MATH · View at MathSciNet
- W. G. Faris, “Inequalities and uncertainty principles,” Journal of Mathematical Physics, vol. 19, no. 2, pp. 461–466, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
- J. F. Price, “Inequalities and local uncertainty principles,” Journal of Mathematical Physics, vol. 24, no. 7, pp. 1711–1714, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. F. Price, “Sharp local uncertainty inequalities,” Studia Mathematica, vol. 85, no. 1, pp. 37–45, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. Fitouhi, N. Bettaibi, and K. Brahim, “The Mellin transform in quantum calculus,” Constructive Approximation, vol. 23, no. 3, pp. 305–323, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. Fitouhi, M. M. Hamza, and F. Bouzeffour, “The Bessel function,” Journal of Approximation Theory, vol. 115, no. 1, pp. 144–166, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet