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Windowed Bessel Fourier transform in quantum calculus and applications. (English) Zbl 1379.33024

Summary: This paper deals firstly with some \(q\)-harmonic analysis properties for the \(q\)-windowed Bessel Fourier transform related to the \(q\)-Bessel function of the third kind as Plancherel formula, inversion formula in \(\mathcal {L}_{q,2,\nu}\). Secondly, we give a weak uncertainty principle for it and we show that the portion of the \(q\)-windowed Bessel Fourier transform lying outside some set of finite measure cannot be arbitrarily too small. Then, we verify a version of Heisenberg-Pauli-Weyl type uncertainty inequalities for the \(q\)-windowed Bessel Fourier transform and its generalization. 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.: Wiley (1988; Zbl 0652.30003)], we will be able to realize the natural and powerful approximation problems that lead to the \(q\)-windowed Bessel Fourier transform inverses.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Citations:

Zbl 0652.30003
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References:

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