On binary B-splines of second order

S. F. Lukomskii ; M. D. Mushko

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 2, pp. 172-182

Résumé

The classical B-spline is defined recursively as the convolution $B_{n+1}=B_n*B_0$, where $B_0$ is the characteristic function of the unit interval. The classical B-spline is a refinable function and satisfies the Riesz inequality. Therefore any B-spline $B_n$ generates the Riesz multiresolution analysis (MRA). We define binary B-splines, obtained by double integration of the third Walsh function. We give an algorithm for constructing an interpolating spline of the second degree for a binary node system and find the approximation order of this interpolation process. We also prove that the system of dilations and shifts of the constructed B-spline generates an MRA $ (V_n) $ in De Boor sense. This MRA is not Riesz. But we can find the approximation order of functions from the Sobolev spaces $W_2^s, s>0$ by the subspaces $ (V_n) $.