Decompositions of Trigonometric Polynomials with Applications to Multivariate Subdivision Schemes

TL;DR

This paper introduces a constructive decomposition method for multivariate trigonometric polynomials, aiding in analyzing convergence and smoothness of multivariate subdivision schemes with general dilation matrices.

Contribution

It develops a simple constructive decomposition technique for multivariate trigonometric polynomials and applies it to establish new sufficient conditions for convergence and smoothness of subdivision schemes.

Findings

Sufficient conditions for convergence applicable to arbitrary dilation matrices.

New criteria for C^1 smoothness of subdivision scheme limits.

Detailed proofs of existing theorems on multivariate matrix subdivision schemes.

Abstract

We study multivariate trigonometric polynomials, satisfying a set of constraints close to the known Strung-Fix conditions. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest to the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. We apply these decompositions, by verifying sufficient conditions for the convergence and smoothness of multivariate scalar subdivision schemes, proved here. For the convergence analysis our sufficient conditions apply to arbitrary dilation matrices, while the previously known necessary and sufficient conditions are relevant only in case of dilation matrices with a self similar tiling. For the analysis of smoothness, we state and prove two theorems on multivariate matrix subdivision schemes, which lead to sufficient conditions for C^1 limits of scalar multivariate subdivision schemes associated with isotropic dilation matrices. Although similar results are stated in the literature, we give here detailed proofs of the results, which we could not find elsewhere.