Fourier series and its property

Abstract

Fourier analysis appeared to solve some partial differential equations at first. In 1747, D’ Alembert solved the vibrating string equation using the method of traveling waves. In 1753, D. Bernoulli proposed the solution which for all intents and purposes is the Fourier series, but Euler did not convince of its full generality entirely since he could not make sure whether any function could be expanded in the Fourier seriesFourier analysis is an important theorem of modern mathematics. It deeply influences the development of partial differential equations and information science. This paper will talk about some basic propositions of the Fourier series and some simple theorems of the Fourier transform. It will use an algebraic view to talk about convolution and good kernels. It will refer to some methods of Hilbert space and complex analysis.