The wavelet transforms in Gelfand-Shilov spaces

TL;DR

This paper investigates the properties of wavelet transforms within Gelfand-Shilov spaces, introducing new localized function spaces and analyzing the wavelet synthesis operator in the context of ultradifferentiable functions.

Contribution

It provides a detailed analysis of wavelet transforms on Gelfand-Shilov spaces, introduces new localized spaces, and studies the wavelet synthesis operator and Calderón reproducing formula for ultradistributions.

Findings

Continuity properties of wavelet transforms on Gelfand-Shilov spaces established.

Introduction of new highly localized time-scale spaces on the upper half-space.

Resolution of identity via wavelet synthesis operator for ultradifferentiable functions achieved.

Abstract

We describe local and global properties of wavelet transforms of ultradifferentiable functions. The results are given in the form of continuity properties of the wavelet transform on Gelfand-Shilov type spaces and their duals. In particular, we introduce a new family of highly time-scale localized spaces on the upper half-space. We study the wavelet synthesis operator (the left-inverse of the wavelet transform) and obtain the resolution of identity (Calder\'{o}n reproducing formula) in the context of ultradistributions.