The Introductory Analysis on Fourier Series

Abstract

Numerical analysis benefits from spectral methods. The Fourier series and Chebyshev polynomials were initially used to solve ordinary differential equations by Cornelius Lanczos. For the first time, spectral approaches were employed by Kreiss, Oliger, Orszag, and other scholars in the 1970s to resolve the partial differential equations of fluid mechanics. In Fourier analysis, general functions are expressed as the sums of a variety of fundamental function options, such as sines and cosines.This paper gives an introductory analysis on Fourier Series. Starting with the basic definition of Fourier coefficients and Fourier Series, the paper introduces the concept and properties of convolution. It also talks about different kernels and good kernel family, analyzing the relationship between these kernels and the original function. Then the paper will combine these ideas and prove the uniqueness and convergence of Fourier Series.