Classical Theory of Fourier Series:Demystified and Generalised

TL;DR

This paper clarifies the classical Fourier series theory, introduces generalizations like finite and one-sided Fourier series, and explores convergence properties for Riemann integrable functions.

Contribution

It extends classical Fourier series theory by defining pointwise Fourier series and introducing new concepts such as finite and one-sided Fourier series.

Findings

Conditions for Fourier series convergence at a point

Introduction of finite and one-sided Fourier series concepts

Summation of subseries in arithmetic progression

Abstract

For a Riemann integrable function on an interval and for a point therein,we define 'Fourier series at the point on the interval' and bring out how and when the function element becomes expressible as Fourier series.In this process,we also generalise the theory by bringing in such concepts as finite Fourier series,right/left hand Fourier series.We also sum up subseries corresponding to terms in an arithmetic progression,of the basic Fourier series.