Gabor transform in quantum calculus and applications. (English) Zbl 1182.33026
Summary: Using the \(q\)-Jackson integral and some elements of the \(q\)-harmonic analysis associated with zero order \(q\)-Bessel operator, for a fixed \(q\in ]0,1[\), 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 \(l_q^2(\mathbb R_{q,+},xd_qx)\) 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 S. Saitoh [“Theory of reproducing kernels and its applications”, Harlow (UK): Longman Scientific & Technical; New York etc.: John Wiley & Sons (1988; Zbl 0652.30003)], we give the \(q\) analogue of the practical real inversion formula for \(q\)-Bessel Gabor transform.
MSC:
33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
42C15 | General harmonic expansions, frames |
44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
33C20 | Generalized hypergeometric series, \({}_pF_q\) |