AUTHOR=Christensen Ole , Goh Say Song 

TITLE=Construction of Scaling Partitions of Unity

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=3

YEAR=2017

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2017.00021

DOI=10.3389/fams.2017.00021

ISSN=2297-4687

ABSTRACT=<p>Partitions of unity in ℝ<sup><italic>d</italic></sup> 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 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.</p>