Fourier decay of absolutely and H\"older continuous functions with infinitely or finitely many oscillations

TL;DR

This paper establishes precise decay rates for Fourier coefficients of functions that are absolutely continuous and H"older continuous, especially those with finitely many oscillations, with applications to numerical derivatives.

Contribution

It provides exact Fourier decay estimates under specific regularity and oscillation conditions, extending to Fourier transforms and applications in numerical analysis.

Findings

Fourier decay rate can be exactly estimated under given conditions.

Decay rates match the uniform H"older continuity of functions.

Applications include error estimation in numerical Weyl fractional derivatives.

Abstract

The main result of this paper is, that if we suppose that a function is absolutely continuous and uniformly H\"older continuous and that its finite difference function does not oscillate infinitely often on a bounded interval, then the decay rate of its Fourier coefficients can be estimated exactly. This rate of decay predicts the same uniform H\"older continuity but the two other conditions are not necessary. Several examples from literature and by the author show that none of the assumptions can be relaxed without weakening the decay for some functions. The uniform H\"older continuity of chirps and the decay of their Fourier coefficients are studied. The main result is then applied in the estimation of the error of numerical Weyl fractional derivatives calculated using the discrete Fourier transform. The main result is also extended to Fourier transforms.