Calderón's reproducing formula and uncertainty principle for the continuous wavelet transform associated with the \(q\)-Bessel operator
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.
scientific article; zbMATH DE number 6939194
scientific article