Adelic multiresolution analysis, construction of wavelet bases and pseudo-differential operators (Q485207)

The paper is a continuation of the article by \textit{A. Y. Khrennikov} et al. [J. Fourier Anal. Appl. 18, No. 6, 1215--1264 (2012; Zbl 1272.42023)], in which an analog of the Haar multiresolution analysis \(\{ V_j\}\) was constructed in \(L^2(\mathbb A)\) where \(\mathbb A\) is the adele ring. Here the authors find explicit orthonormal wavelet bases modifying the above construction. These wavelet functions are shown to be eigenfunctions of the fractional differentiation operator on adeles introduced by \textit{S. M. Torba} and \textit{W. A. Zúñiga-Galindo} [J. Fourier Anal. Appl. 19, No. 4, 792--835 (2013; Zbl 1333.35126)]. See also \textit{S. Evdokimov} [J. Math. Sci., New York 192, No. 2, 215--219 (2013); translation from Zap. Nauchn. Sem. POMI 400, 158--165 (2012; Zbl 1302.11091)].