Wavelet Analysis on Adeles and Pseudo-Differential Operators

This paper is devoted to wavelet analysis on the adele ring and the theory of pseudo-differential operators. We develop the technique which gives the possibility to generalize finite-dimensional results of wavelet analysis to the case of adeles by using infinite tensor products of Hilbert spaces. We prove that is the infinite tensor product of the spaces L (2)(a"e (p) ). This description allows us to construct an infinite family of Haar wavelet bases on which can be obtained by shifts and multi-dilations. The adelic Haar multiresolution analysis (MRA) in is constructed. To do this, we generalize the idea of constructing the separable multidimensional MRA (suggested by Y. Meyer and S. Mallat) to the infinite dimensional case. In the framework of this MRA infinite family of Haar wavelet bases is constructed. We introduce the adelic Lizorkin spaces of test functions and distributions and give the characterization of these spaces in terms of adelic wavelet functions. One class of pseudo-differential operators (including the fractional operator) is studied on the Lizorkin spaces. A criterion for an adelic wavelet function to be an eigenfunction for a pseudo-differential operator is derived. We prove that any wavelet function is an eigenfunction of the fractional operator.