  This paper is devoted to wavelet analysis on adele ring $\bA$ 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 $\bA$ by using infinite tensor products of Hilbert spaces. The
adele ring is roughly speaking a subring of the direct product of all possible
($p$-adic and Archimedean) completions $\bQ_p$ of the field of rational numbers
$\bQ$ with some conditions at infinity. Using our technique, we prove that
$L^2(\bA)=\otimes_{e,p\in\{\infty,2,3,5,...}}L^2({\bQ}_{p})$ is the infinite
tensor product of the spaces $L^2({\bQ}_{p})$ with a stabilization $e=(e_p)_p$,
where $e_p(x)=\Omega(|x|_p)\in L^2({\bQ}_{p})$, and $\Omega$ is a
characteristic function of the unit interval $[0,\,1]$, $\bQ_p$ is the field of
$p$-adic numbers, $p=2,3,5,...$; $\bQ_\infty=\bR$. This description allows us
to construct an infinite family of Haar wavelet bases on $L^2(\bA)$ which can
be obtained by shifts and multi-delations. The adelic multiresolution analysis
(MRA) in $L^2(\bA)$ is also constructed. In the framework of this MRA another
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 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. These results
allow one to create the necessary prerequisites for intensive using of adelic
wavelet bases and pseudo-differential operators in applications.
