6 - Orthogonal Systems

Abstract

This chapter summarizes the Fourier series, Parseval's identity and orthogonal systems whose elements are the eigenfunctions of some Sturm–Liouville problem. It concentrates on the result in the more general setting in which the orthogonal system consists of the eigenfunctions of a regular Sturm–Liouville problem. These orthogonal functions are called the Chebyshev polynomials, because they were introduced by Pafnuti L'vovich Chebyshev, a professor at the University of Saint Petersburg, in answer to the problem related to the polynomials. The chapter also highlights the condition that the completeness of the orthogonal system is a necessary for the convergence of Fourier series in the general case. In view of the established analogy, it is reasonable to conjecture that an orthogonal system is complete if and only if Parseval's identity holds for all integrable functions with respect to that system. It can prove in an easy manner if part in the appropriate context.