On a \(q\)-Paley–Wiener theorem. (English) Zbl 1069.33016
Summary: After the analysis of the \(q\)-even translation and the \(q\)-cosine Fourier transform by A. Fitouhi and F. Bouzeffour in [\(q\)-cosine Fourier transform and \(q\)-heat equation, Ramanujan J., in press], it is natural to look for the \(q\)-analogue of some well-known classical theorems. In this paper, we purpose to give a \(q\)-version of the Paley–Wiener theorem related to this Fourier transform using standard methods of the \(q\)-calculus.
MSC:
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
| 39A13 | Difference equations, scaling (\(q\)-differences) |
References:
| [1] | Carnovale, G.; Koornwinder, T. H., A \(q\)-analogue of convolution on the line, Methods Appl. Anal., 7, 705-726 (2000) · Zbl 1004.33013 |
| [2] | A. Fitouhi, F. Bouzeffour, \(qq\); A. Fitouhi, F. Bouzeffour, \(qq\) · Zbl 1255.33009 |
| [3] | Fitouhi, A.; Hamza, M. M.; Bouzeffour, F., The \(q-j_α\) Bessel function, J. Approx. Theory, 115, 144-166 (2002) · Zbl 1003.33007 |
| [4] | Gasper, G.; Rahman, M., Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, vol. 35 (1990), Cambridge Univ. Press · Zbl 0695.33001 |
| [5] | Jackson, F. H., On \(q\)-functions and a certain difference operator, Trans. Roy. Soc. London, 46, 253-281 (1908) |
| [6] | Jackson, F. H., On a \(q\)-definite integrals, Quart. J. Pure Appl. Math., 41, 193-203 (1910) · JFM 41.0317.04 |
| [7] | T.H. Koornwinder, \(q\); T.H. Koornwinder, \(q\) |
| [8] | Koornwinder, T. H.; Swarttouw, R. F., On \(q\)-analogues of the Hankel and Fourier transform, Trans. Amer. Math. Soc., 333, 445-461 (1992) · Zbl 0759.33007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
