Construction of Scaling Partitions of Unity

Ole Christensen, Say Song Goh
2017 Frontiers in Applied Mathematics and Statistics  
Partitions of unity in 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 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
more » ... 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.
doi:10.3389/fams.2017.00021 fatcat:vtnxf5zz2jgapl3u7t7zjckz4u