GABOR TRANSFORM IN QUANTUM CALCULUS AND APPLICATIONS

Abstract

Abstract In this work, using the q-Jackson integral and some elements of the qharmonic analysis associated with zero order q-Bessel operator, for a fixed q ∈]0, 1[, we study the q analogue of the continuous Gabor transform associated with the q-Bessel operator of order zero. We give some q-harmonic analysis properties (a Plancherel formula, an L 2 q (R q,+ , xd q x) inversion formula, etc), and a weak uncertainty principle for it. Then, we show that the portion of the q-Bessel Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Finally, using the kernel reproducing theory, given by Saitoh [13], we give the q analogue of the practical real inversion formula for q-Bessel Gabor transform. Mathematics Subject Classification: 33D15, 42C15 (main), 44A15, 33

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