Orthogonal Polynomials and Related Approximation Results

Abstract

The Fourier spectral method is only appropriate for problems with periodic boundary conditions. If a Fourier method is applied to a non-periodic problem, it inevitably induces the so-called Gibbs phenomenon, and reduces the global convergence rate to O(N −1) (cf. Gottlieb and Orszag (1977)). Consequently, one should not apply a Fourier method to problems with non-periodic boundary conditions. Instead, one should use orthogonal polynomials which are eigenfunc-tions of some singular Sturm-Liouville problems. The commonly used orthogonal polynomials include the Legendre, Chebyshev, Hermite and Laguerre polynomials. The aim of this chapter is to present essential properties and fundamental approximation results related to orthogonal polynomials. These results serve as preparations for polynomial-based spectral methods in the forthcoming chapters. This chapter is organized as follows. In the first section, we present relevant properties of general orthogonal polynomials, and set up a general framework for the study of orthogonal polynomials. We then study the Jacobi polynomials in Sect. 3.2, Legendre polynomials in Sect. 3.3 and Chebyshev polynomials in Sect. 3.4. In Sect. 3.5, we present some general approximation results related to these families of orthogonal polynomials. We refer to Szegö (1975), Davis and Rabinowitz (1984) and Gautschi (2004) for other aspects of orthogonal polynomials. 3.1 Orthogonal Polynomials Orthogonal polynomials play the most important role in spectral methods, so it is necessary to have a thorough study of their relevant properties. Our starting point is the generation of orthogonal polynomials by a three-term recurrence relation, which leads to some very useful formulas such as the Christoffel-Darboux formula. We then review some results on zeros of orthogonal polynomials, and present efficient algorithms for their computations. We also devote several sections to discussing some important topics such as Gauss-type quadrature formulas, polynomial interpolations , discrete transforms, and spectral differentiation techniques.