Smoothness of Nonlinear and Non-Separable Subdivision Schemes

TL;DR

This paper analyzes the convergence and regularity of nonlinear, multivariate subdivision schemes with arbitrary dilation matrices, providing conditions for their limit functions in various function spaces.

Contribution

It introduces new convergence criteria for nonlinear subdivision schemes in multivariate settings with arbitrary dilation matrices.

Findings

Established convergence conditions based on contractivity of difference schemes.

Proved regularity of limit functions in $L^p$ and Sobolev spaces.

Extended analysis to nonlinear, non-separable subdivision schemes.

Abstract

We study in this paper nonlinear subdivision schemes in a multivariate setting allowing arbitrary dilation matrix. We investigate the convergence of such iterative process to some limit function. Our analysis is based on some conditions on the contractivity of the associated scheme for the differences. In particular, we show the regularity of the limit function, in $L^p$ and Sobolev spaces.