The Weiestrass theorem says that any continuous function on a finite closed interval can be uniformly approximated, with any required accuracy, by polynomials. The Stone–Weierstrass theorem extends this result to an abstract setting, where the interval is replaced by a compact topological space, and the role of polynomials is played by a class of functions that enjoy certain properties mimicking those of polynomials. There are scores of proofs of the latter result; the one presented in this little book could not fail to stress the importance of completeness of the space of continuous functions.

This chapter is a close companion to the previous one. Here we study the best least squares approximation to periodic functions via trigonometric polynomials. Many of the ideas and results of the previous chapter are repeated in this scenario. They are then expanded to deal with merely square integrable functions. The Fourier transform of periodic functions, and its inverse, is then introduced and studied. Uniform convergence of trigonometric series, under several different smoothness assumptions is then discussed.Trigonometric approximation in periodic Sobolev spaces is then discussed.