Fourier Analysis and Its Application

Abstract

In this article, mathematicianswill give a general summary of Fourier analysis and some of its applications. The Fourier transform will be organized in a developing order, the discrete Fourier transform, Fourier transform on the unit circle and Fourier transform on the real line. Some theorems about the convergence of Fourier series in different forms will be proved in detail. Finally, an estimation of the upper bound of Fourier transform will be discussed. Mathematicianscan prove that it is controlled by the 1-norm of the derivative of f.