Calderón’s reproducing formula and uncertainty principle for the continuous wavelet transform associated with the \(q\)-Bessel operator. (English) Zbl 1402.42007
The paper falls into the scope of the \(q\)-harmonic analysis associated to the \(q\)-Bessel operator
\[
\Delta_{q,\nu} f(x) =\frac{f(q^{-1}x)-(1+q^{2\nu}) f(x) + q^{2\nu} f(qx)}{x^2}, \qquad x\neq 0.
\]
This analysis includes the associated \(q\)-Bessel Fourier transform, the \(q\)-Bessel translation operator, and the qconvolution product. The authors introduce the concepts of \(q\)-wavelet and continuous \(q\)-wavelet transform and prove the corresponding analogue of Calderón’s reproducing formula in the \(L^2\) setting. The uncertainty principle for these transforms is also studied.
Reviewer: Boris Rubin (Baton Rouge)
MSC:
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 44A20 | Integral transforms of special functions |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 33B15 | Gamma, beta and polygamma functions |
| 33D05 | \(q\)-gamma functions, \(q\)-beta functions and integrals |
Keywords:
\(q\)-wavelet; continuous \(q\)-wavelet transform; \(q\)-Bessel operator; \(q\)-Bessel Fourier transform; Calderón’s reproducing formula; uncertainty principle; Heisenberg-Pauli-Weyl inequalityReferences:
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