Mapping properties of Fourier transforms

TL;DR

This paper investigates the properties of Fourier transforms as compact mappings between specific function spaces, focusing on entropy numbers to measure their degree of compactness and emphasizing the interplay of known mathematical tools.

Contribution

It provides a detailed analysis of the compactness properties of Fourier transforms between function spaces, using entropy numbers as a key measure, with an emphasis on existing mathematical ingredients.

Findings

Fourier transforms act as compact mappings in certain function spaces.

Entropy numbers effectively measure the degree of compactness.

The study highlights the interplay of known mathematical tools in analyzing Fourier transform properties.

Abstract

The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished function spaces on $\mathbb{R}^n$. The degree of compactness will be measured in terms of related entropy numbers. We are more interested in the interplay of already available ingredients than in generality.