Special Weber Transform with Nontrivial Kernel

Abstract: We study the Weber integral transforms <inline-formula id="IEq1"><alternatives><tex-math id="IEq1_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{k,k\pm1}$$\end{document}</tex-math><inline-graphic href="11006_2023_2661_Article_IEq1.gif"></inline-graphic></alternatives></inline-formula>, which have a nontrivial kernel, so that the spectral expansion contains not only the continuous part of the spectrum but also the zero eigenvalue corresponding to the kernel. The inversion formula, the spectral decomposition, and the Plancherel–Parseval equality are derived. These transforms are used in an explicit formula for the solution of the classical nonstationary Stokes problem on the flow past a circular cylinder.