Limits of level and parameter dependent subdivision schemes: a matrix approach

TL;DR

This paper introduces a matrix-based method to analyze the convergence and regularity of level and parameter dependent subdivision schemes, highlighting their limitations and improvements over stationary schemes.

Contribution

It presents a novel matrix approach for analyzing non-stationary subdivision schemes with parameter dependency, enabling efficient convergence and regularity checks.

Findings

The method effectively determines convergence and Hölder regularity.

Parameter dependency can enhance subdivision scheme properties.

Necessary criteria for functions generated by level-dependent schemes are established.

Abstract

In this paper, we present a new matrix approach for the analysis of subdivision schemes whose non-stationarity is due to linear dependency on parameters whose values vary in a compact set. Indeed, we show how to check the convergence in $C^{\ell}(\RR^s)$ and determine the H\"older regularity of such level and parameter dependent schemes efficiently via the joint spectral radius approach. The efficiency of this method and the important role of the parameter dependency are demonstrated on several examples of subdivision schemes whose properties improve the properties of the corresponding stationary schemes. Moreover, we derive necessary criteria for a function to be generated by some level dependent scheme and, thus, expose the limitations of such schemes.