Multiresolution analysis in the space $\ell^2(\mathbb Z)$ using discrete splines

A non-stationary multiresolution analysis $\{V_k\}_{k\geq 0}$ $\ell^2(\mathbb Z)$ in the space $\ell^2(\mathbb Z)$ is performed, the subspaces $V_k$ consisting of discrete splines. In each $V_k$, there is a function $\varphi_k$ such that the system $\{\varphi_k(\cdot-l2^k):l\in\mathbb Z\}$ forms the Riesz base of $V_k$. A system of wavelets $\psi_{kl}(j)=\psi_k(j-l2^k)$, $l\in\mathbb Z$, $k=1,2\dots$ is not generated by shifts and dilations of the unique function. The subspaces $W_k=\operatorname{span}\{\psi_{kl}:l\in\mathbb Z\}$ form an orthogonal expansion of the space: $\ell^2(\mathbb Z)=\oplus^{\infty}_{k=1}W_k$.
The space $V_k$ is the same as the space of discrete splines $S_{p,2^k}$ of order $p$ with a distance between the knots $2^k$. For every $p$, a multiresolution analysis is obtained (for $p=1$ – the Haar multiresolution analysis).