Construction of scaling partitions of unity

TL;DR

This paper characterizes functions and matrices that generate partitions of unity in br^d, providing methods to construct such functions with high regularity and small support, with applications to wavelet frames and nonuniform splines.

Contribution

It offers a simple characterization of functions and matrices forming partitions of unity, enabling new constructions with desirable properties and applications in harmonic analysis.

Findings

Characterization of functions and matrices for partitions of unity

Construction methods for functions with high regularity and small support

Application to wavelet frames and nonuniform splines

Abstract

Partitions of unity in ${\mathbf R}^d$ formed by (matrix) scales of a fixed function appear in many parts of harmonic analysis, e.g., wavelet analysis and the analysis of Triebel-Lizorkin spaces. We give a simple characterization of the functions and matrices yielding such a partition of unity. For invertible expanding matrices, the characterization leads to easy ways of constructing appropriate functions with attractive properties like high regularity and small support. We also discuss a class of integral transforms that map functions having the partition of unity property to functions with the same property. The one-dimensional version of the transform allows a direct definition of a class of nonuniform splines with properties that are parallel to those of the classical B-splines. The results are illustrated with the construction of dual pairs of wavelet frames.