Multivariate tile $\mathrm{B}$-splines

Abstract

Tile $\mathrm{B}$-splines in $\mathbb R^d$ are defined as autoconvolutions of indicators of tiles, which are special self-similar compact sets whose integer translates tile the space $\mathbb R^d$. These functions are not piecewise-polynomial, however, being direct generalizations of the classical $\mathrm{B}$-splines, they enjoy many of their properties and have some advantages. In particular, exact values of the Hölder exponents of tile $\mathrm{B}$-splines are evaluated and are shown, in some cases, to exceed those of the classical $\mathrm{B}$-splines. Orthonormal systems of wavelets based on tile B-splines are constructed, and estimates of their exponential decay are obtained. Efficiency in applications of tile $\mathrm{B}$-splines is demonstrated on an example of subdivision schemes of surfaces. This efficiency is achieved due to high regularity, fast convergence, and small number of coefficients in the corresponding refinement equation.