Multivariate tile B-splines

TL;DR

This paper introduces multivariate tile B-splines, explores their properties, computes their Hölder exponents, constructs wavelet systems, and demonstrates their efficiency in subdivision schemes for applications.

Contribution

It generalizes classical B-splines using tile-based autoconvolutions, computes their regularity, and develops wavelet systems with proven decay properties.

Findings

Hölder exponents sometimes exceed classical B-splines

Orthogonal wavelet systems with exponential decay are constructed

Subdivision schemes show high efficiency due to regularity and convergence

Abstract

Tile B-splines in $\mathbb{R}^d$ are defined as autoconvolutions of the 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 classical B-splines, they enjoy many of their properties and have some advantages. In particular, the precise values of the H\"older exponents of the tile B-splines are computed in this work. They sometimes exceed the regularity of the classical B-splines. The orthonormal systems of wavelets based on the tile B-splines are constructed and the estimates of their exponentional decay are obtained. Subdivision schemes constructed by the tile B-splines demonstrate their efficiency in applications. It is achieved by means of the high regularity, the fast convergence, and small number of the coefficients in the corresponding refinement equation.