Spectral accuracy for the Hahn polynomials

Abstract

We consider in this paper the Hahn polynomials and their application in numerical methods. The Hahn polynomials are classical discrete orthogonal polynomials. We analyse the behaviour of these polynomials in the context of spectral approximation of partial differential equations. We study series expansions $u=\sum_{n=0}^\infty \hat{u}_n ϕ_n$, where the $ϕ_n$ are the Hahn polynomials. We examine the Hahn coefficients and proof spectral accuracy in some sense. We substantiate our results by numericals tests. Furthermore we discuss a problem which arise by using the Hahn polynomials in the approximation of a function $u$, which is linked to the Runge phenomenon. We suggest two approaches to avoid this problem. These will also be the motivation and the outlook of further research in the application of discrete orthogonal polynomials in a spectral method for the numerical solution of hyperbolic conservation laws.