Orthogonal Polynomials and Weighted Polynomial Approximation

Abstract

AbstractThe second chapter is devoted to orthogonal polynomials on the real line and weighted polynomial approximation. In Sect. 2.1 we introduce the concept of orthogonality and consider several examples of orthogonal systems. In particular, we study Fourier expansions and best approximation. In Sect. 2.2 we give the basic properties of polynomials orthogonal on the real line and introduce and discuss the associated polynomials, functions of the second kind, Stieltjes polynomials, as well as the Christoffel functions and numbers. The classical orthogonal polynomials as the most important class of orthogonal polynomials on the real line are treated in Sect. 2.3, and new results on nonclassical orthogonal polynomials, including methods for their numerical construction, are studied in Sect. 2.4. Introducing the weighted functional spaces, moduli of smoothness and K-functionals, the weighted best polynomial approximations on (−1, 1), (0,+∞) and (−∞, +∞) are treated in Sect. 2.5, as well as the weighted polynomial approximation of functions having interior isolated singularities.Mathematics Subject Classification (2000)33-xx41-xx42Axx45A0545B0545H0565B1065Dxx