Approximation of Periodic Functions

Abstract

In the study of approximation of functions in Chaps. 1—8, the emphasis is on (algebraic) polynomial approximation on the bounded interval $[a,b]$. Since algebraic polynomials are not periodic functions, they are not suitable basis functions for representing and approximating periodic functions on the entire real line ${\text{R}}$. On the other hand, many natural phenomena can only be represented by periodic functions. It is therefore essential to study approximation of periodic continuous functions $f:{\text{R}} \to {\text{R}}$, by the linear span of some elementary periodic functions. This chapter is devoted to the study of this topic by considering basis functions that are formulated in terms of the sine and cosine functions.