V - Orthogonal Polynomials

Abstract

This chapter focuses on simple sets of orthogonal polynomials. These sets of polynomials arise in various ways, one of which is as the solutions of a class of differential equations. It has been shown that, under certain conditions, given any interval and a positive weight function on that interval, there exists a corresponding set of orthogonal polynomials. Eigenvalue problems are another important source of sets of orthogonal functions, not necessarily polynomials. It has also been shown that under certain conditions, an arbitrary function could be expanded in an infinite series of the functions of an orthogonal set. Such series are called Fourier series. The knowledge of Fourier series and orthogonal functions can be used to obtain the solutions of some problems of mathematical physics. Some basic general properties of sets of orthogonal polynomials that hold for all such sets have been derived in the chapter.