Convergence and divergence of wavelet series: multifractal aspects

TL;DR

This paper investigates the multifractal properties of wavelet series, providing bounds on the dimensions of convergence and divergence sets, and demonstrating their optimality in a generic sense.

Contribution

It introduces bounds on Hausdorff and packing dimensions for convergence/divergence points of wavelet expansions in Sobolev and Besov spaces, establishing their optimality.

Findings

Bounds on Hausdorff and packing dimensions are derived.

These bounds are shown to be generically optimal.

The results connect wavelet convergence behavior with multifractal analysis.

Abstract

We study the convergence and divergence of the wavelet expansion of a function in a Sobolev or a Besov space from a multifractal point of view. In particular, we give an upper bound for the Hausdorff and for the packing dimension of the set of points where the expansion converges (or diverges) at a given speed, and we show that, generically, these bounds are optimal.