Fourier series for nonperiodic functions

TL;DR

This paper proposes a modified Fourier series definition that ensures endpoint accuracy for nonperiodic functions, eliminates Gibbs phenomenon, and aids in solving antiperiodic boundary value problems in PDEs.

Contribution

It introduces a new Fourier series formulation that aligns with nonperiodic functions at endpoints and improves convergence properties.

Findings

Guarantees endpoint coincidence for nonperiodic functions

Eliminates Gibbs phenomenon at interval endpoints

Facilitates solving antiperiodic boundary value problems

Abstract

We introduce a small change in the definition of the Fourier series so that we can guarantee the coincidence with the given function at the endpoints of the interval even if the function does not assume the same value at the endpoints. This definition of the Fourier series also wipes out the Gibbs phenomenom at the endpoints of the interval and proves useful in the resolution of antiperiodic boundary value problems with linear partial differential equations.