Binary basic splines in MRA

$B$-splines were introduced by Carry and Schoenberg. Constructed on a uniform mesh and defined in terms of convolutions, such splines generate a Riesz MRA. We constructed splines $\varphi_n$, where $n$ is the order of integration of the Walsh function with the number $2^n - 1$. We called these splines binary basic splines. We know that binary basic splines form a basis in the space of functions that are continuous on the segment $[0, 1]$ and $0$ outside of it. We proved that binary basic splines are a scaling function and generate an MRA of $(V_n)$ which is not a Riesz MRA. The order of approximation was determined by subspaces from Sobolev spaces.