ABSTRACT
In this chapter we extend and refine the introduction to trigonometric Fourier series begun in Chapter 1. Stronger pointwise convergence results are presented here, as well as rates of convergence. In addition, several new topics are taken up. These include Gibbs phenomenon which involves overshoot at discontinuities and two types of summability, Cesaro and Abel, used to overcome i t . We then consider periodic tempered distributions and use them for a more general theory of Fourier series in which the distinction between formal trigonometric series and Fourier series is eliminated.
