Estimations of Fourier Coefficients For Controlled Distortion of a Periodic Function

TL;DR

This paper investigates how to estimate Fourier coefficients to control the distortion of periodic functions, with applications in physical filtration and sound generation, establishing inequalities relating function characteristics to Fourier coefficients.

Contribution

It introduces methods for estimating Fourier coefficients to achieve controlled distortion of periodic functions, linking integral characteristics to harmonic amplitudes.

Findings

Derived inequalities relating Fourier coefficients to function characteristics

Demonstrated control of harmonic amplitudes in physical filtration processes

Applied results to sound generation by revolving bodies in fluid flow

Abstract

There is a class of physical filtration processes where the input is adequately modeled by a continuous periodic function f (x) of bounded variation over its period, and the output depends only on certain harmonics of the Fourier expansion of f (x) in the orthogonal basis of trigonometric functions. One example is the discrete spectrum sound generation by a revolving body in a steady fluid flow. This type of sound can be controlled through the amplitudes of certain harmonics of the circular distribution of the inflow velocity. Attainable goals of such a controlled distortion of f (x) are formulated as fuzzy targets and required inequalities relating the integral characteristics of f (x) with the coefficients of the corresponding Fourier expansion are proven.