Multivariate compactly supported $C^\infty$ functions by subdivision

TL;DR

This paper introduces a method to generate multivariate $C^ty$ functions with small compact supports using subdivision schemes, extending univariate constructions and enabling potential wavelet applications.

Contribution

It develops new tools for analyzing non-stationary subdivision schemes, producing Up-like functions with smaller supports than previous methods.

Findings

Univariate Up-like functions with supports $[0, 1 + psilon]$

Construction of bivariate Up-like functions

Potential for generating small support $C^ty$ wavelets

Abstract

This paper discusses the generation of multivariate $C^\infty$ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called \emph{Up-function}, by a non-stationary scheme based on masks of {spline subdivision schemes} of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes than the class we study. With our method for achieving small compact supports, we obtain, in the univariate case, Up-like functions with supports $[0, 1 +\epsilon ]$ in comparison to the support $[0, 2] $ of the Up-function. Examples of univariate and bivariate Up-like functions are given. As in the univariate case, the construction of Up-like functions can motivate the generation of $C^\infty$ compactly supported wavelets of small support in any dimension.