# Multiresolution expansions and wavelets in Gelfand-Shilov spaces

## TL;DR

This paper investigates the approximation and convergence properties of wavelet expansions within Gelfand-Shilov spaces, focusing on highly regular wavelets with subexponential decay, and extends understanding of multiresolution analysis in these spaces.

## Contribution

It provides new results on approximation and convergence of wavelet series in Gelfand-Shilov spaces, especially for band-limited wavelets with subexponential decay.

## Key findings

- Approximation properties of multiresolution expansions for Gelfand-Shilov functions.
- Convergence of wavelet series in Gelfand-Shilov spaces.
- Application to band-limited wavelets with subexponential decay.

## Abstract

We study approximation properties generated by highly regular scaling functions and orthonormal wavelets. These properties are conveniently described in the framework of Gelfand-Shilov spaces. Important examples of multiresolution analyses for which our results apply arise in particular from Dziuba\'{n}ski-Hern\'{a}ndez construction of band-limited wavelets with subexponential decay. Our results are twofold. Firstly, we obtain approximation properties of multiresolution expansions of Gelfand-Shilov functions and (ultra)distributions. Secondly, we establish convergence of wavelet series expansions in the same regularity framework.