Breve abordagem às séries de fourier e suas aplicaçoes

Abstract

With this work we intend to approach, in a simple way, the theory of Fourier series and its application in the resolution of several problems in mathematical physics. In the first chapter, a digression is made to the Fourier series. Some theorems are formulated and demonstrated, such as the Cauchy-Bunikowskii inequality theorem, Dini's theorem on punctual convergence of partial sums of Fourier series and the theorem on the uniform convergence of a Fourier series. The notion of an orthogonal system and a linearly independent system is introduced. Due to its relevance, a quick review of the Euclidean L2 space is made. An exhaustive study is made of the Cesàro sums and the Fejér nucleus. It is shown that if f(x) is a continuous, 27r-periodic function, then the Cesàro sum, generated by /(x), converges uniformly to f(x). Some orthogonal systems are studied for the case of two variables and the decomposition of functions into double Fourier series. In the second chapter a brief approach to the Euler functions of the first kind is made, the nucleus and sum of Vallée Poussin are investigated. A study of the integral of the Vallée Poussin nucleus is made and an evaluation of the type — In (1 4- - ) +1.7 is obtained. Finally, in the last chapter, the method of eigenfunctions is studied. The Sturm-Liouville operator is shown to be symmetrical and positive. When dealing, in the applications of the method of proper functions, only with problems of oscillation and heat transmission, it was assumed that the application of the theory should focus, in the first moments, on phenomena that are simple to understand. Using the Maple® package, Fourier series generated by certain functions are built and graphic illustrations are made where exact functions and their approximations are compared.