Change of variable and the rapidity of decrease of Fourier coefficients

TL;DR

This paper explores how changing variables can influence the rate at which Fourier coefficients of continuous functions on the circle decrease, aiming to classify functions based on this property.

Contribution

It investigates the possibility of transforming functions into classes characterized by Fourier coefficient decay via variable changes.

Findings

Identifies classes of functions where Fourier coefficient decay can be accelerated

Provides conditions under which variable changes affect Fourier decay rates

Offers a corrected translation of the original Russian publication

Abstract

We consider the class $C(T)$ of continuous real-valued functions on the circle. For certain classes of functions naturally characterised by the rapidity of decrease of Fourier coefficients we investigate whether it is possible to bring families of functions in $C(T)$ into these classes by a change of variable. This paper was originally published in Matematicheski\v{\i} Sbornik, 181:8 (1990), 1099--1113 (Russian). The English translation, published in Mathematics of the USSR, Sbornik, 70:2 (1991), 541--555, is to a large extent inconsistent with the original text. Herein the author provides a corrected translation.