Fourier series

Abstract

Chapter 1 covers the classical theory of Fourier series of 2π-periodic functions. In the introductory section, we sketch Fourier’s theory on heat propagation. Section 1.2 introduces some basic notions such as Fourier coefficients and Fourier series of a 2π-periodic function. The convolution of 2π-periodic functions is handled in Sect. 1.3. Section 1.4 presents main results on the pointwise and uniform convergence of Fourier series. For a 2π-periodic, piecewise continuously differentiable function f, a complete proof of the important convergence theorem of Dirichlet–Jordan is given. Further we describe the Gibbs phenomenon for partial sums of the Fourier series of f near a jump discontinuity. Finally, in Sect. 1.5, we apply Fourier series in digital signal processing and describe the linear filtering of discrete signals.