Multiresolution expansions and wavelets in Gelfand–Shilov spaces

Abstract

© 2020, The Royal Academy of Sciences, Madrid. 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 Dziubański–Hernández 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.