Wavelet-type Transform and Bessel Potentials Associated with the Generalized Translation

Wavelet–type transform associated with singular Laplace–Bessel differential operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta _\nu = \sum\limits_{k = 1}^n {\frac{{\partial ^2 }} {{\partial x_k^2 }}} + \frac{{2\nu}} {{x_n }}\frac{\partial } {{\partial x_n }}$ \end{document} is introduced and the relevant Calderón–type reproducing formula is established. Representations of the generalized Bessel potentials \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(E - \Delta _\nu )^{ - \alpha /2} f,\quad (Re \alpha > 0)$ \end{document} and their inverses via the wavelet–type transform are obtained.