# Regularity of anisotropic refinable functions

## TL;DR

This paper develops a method to precisely compute the H"older regularity of anisotropic refinable functions, extending previous univariate and isotropic results to the more complex anisotropic multivariate setting.

## Contribution

It introduces an exact formula for H"older regularity in anisotropic cases and provides an efficient algorithm for its computation, advancing understanding of multivariate wavelet regularity.

## Key findings

- Exact H"older exponent formula for anisotropic refinable functions
- Efficient algorithm for spectral analysis of transition matrices
- Analysis of higher and local regularity, and convergence rates

## Abstract

This paper presents a detailed regularity analysis of anisotropic wavelet frames and subdivision. In the univariate setting, the smoothness of wavelet frames and subdivision is well understood by means of the matrix approach. In the multivariate setting, this approach has been extended only to the special case of isotropic refinement with the dilation matrix all of whose eigenvalues are equal in the absolute value. The general anisotropic case has resisted to be fully understood: the matrix approach can determine whether a refinable function belongs to $C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p < \infty$, but its H\"older regularity remained mysteriously unattainable. It this paper we show how to compute the H\"older regularity in $C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p < \infty$. In the anisotropic case, our expression for the exact H\"older exponent of a refinable function reflects the impact of the variable moduli of the eigenvalues of the corresponding dilation matrix. In the isotropic case, our results reduce to the well-known facts from the literature. We provide an efficient algorithm for determining the finite set of the restricted transition matrices whose spectral properties characterize the H\"older exponent of the corresponding refinable function. We also analyze the higher regularity, the local regularity, the corresponding moduli of continuity, and the rate of convergence of the corresponding subdivision schemes. We illustrate our results with several examples.