Optimal H\"older-Zygmund exponent of semi-regular refinable functions

TL;DR

This paper introduces an efficient wavelet-based method to estimate the optimal H"older-Zygmund regularity of semi-regular refinable functions, overcoming limitations of previous Fourier-based approaches.

Contribution

It develops a new wavelet tight frame characterization for irregular settings and proposes a more efficient numerical approach for regularity estimation.

Findings

The method accurately estimates regularity of semi-regular subdivision schemes.

It outperforms traditional linear regression methods in efficiency.

Applications include blending curve design with semi-regular meshes.

Abstract

The regularity of refinable functions has been investigated deeply in the past 25 years using Fourier analysis, wavelet analysis, restricted and joint spectral radii techniques. However the shift-invariance of the underlying regular setting is crucial for these approaches. We propose an efficient method based on wavelet tight frame decomposition techniques for estimating H\"older-Zygmund regularity of univariate semi-regular refinable functions generated, e.g., by subdivision schemes defined on semi-regular meshes $\mathbf{t}\;=\;-h_\ell\mathbb{N}\cup\{0\}\cup h_r\mathbb{N}$, $h_\ell,h_r \in (0,\infty)$. To ensure the optimality of this method, we provide a new characterization of H\"older-Zygmund spaces based on suitable irregular wavelet tight frames. Furthermore, we present proper tools for computing the corresponding frame coefficients in the semi-regular setting. We also propose a new numerical approach for estimating the optimal H\"older-Zygmund exponent of refinable functions which is more efficient than the linear regression method. We illustrate our results with several examples of known and new semi-regular subdivision schemes with a potential use in blending curve design.