Haar multiresolution analysis and Haar bases on the ring of rational adeles

We construct a family of Haar multiresolution analyses in the Hilbert space $L^2(\mathbb A)$ where $\mathbb A$ is the ring of adeles over the field $\mathbb Q$ of rationals. The corresponding discrete group of translations and scaling function are respectively the group of additive translations by elements of $\mathbb Q$ embedded diagonally in $\mathbb A$ and the characteristic function of the standard fundamental domain of this group. As a consequence we come to a family of orthonormal wavelet bases in $L^2(\mathbb A)$. We observe that both the number of generating wavelet functions and the number of elementary dilations are infinite.