Fourier Series and Integrals

ABSTRACT: This thesis consists of six chapters. Introduction is in the first chapter. In the second chapter we present a method for solving partial differential equation by use of Fourier series. The method is called separation of variables. In the third chapter we show that the Fourier series converges under certain reasonable general hypothesis. We give important results like Riemann-Lebesgue Lemma, Dirichlet kernels and three important conditions for the convergence of Fourier series at a point Dini’s, Lipchitz and Dirichlet-Jordan conditions. In the fourth chapter Fourier series are studied in more general point of view, considering functions as elements of abstract inner product space. Bessel’s inequality, Parseval’s identity, Cesaro summability and Fejer kernels are important results that are given. In the fifth chapter is set the problem of uniform convergence of Fourier series based on piecewise-smooth functions. In addition it is given Weierstrass approximation theorem and Gibbs phenomenon, the case when the function is not uniformly convergent. In the last chapter we deal with convergence of Fourier integrals. First we introduce the Fourier integral formula and then give the analogs of Dini’s, Lipchitz and Dirichlet- Jordan conditions for Fourier integrals. Keywords: Dirichlet kernels, Bessel’s inequality, Parseval’s identity, Cesaro summability, Fejer kernels. ……………………………………………………………………………………………………………………………………………………………………………………………………………………