Discrete (Legendre) orthogonal polynomials—a survey

Abstract

Abstract The discrete (Legendre) orthogonal polynomials , (DLOP's) are useful for approximation purposes. This set of m th degree polynomials { P m ( K , N )} are orthogonal with unity weight over a uniform discrete interval and are completely determined by the normalization P m (O, N ) = 1. The authors are employing these polynomials as assumed modes in engineering applications of weighted residual methods. Since extensive material on these discrete orthogonal polynomials, and their properties, is not readily available, this paper is designed to unify and summarize the presently available information on the DLOP's and related polynomials. In so doing, many new properties have been derived. These properties, along with sketches of their derivation, are included. Also presented are a representation of the DLOP's as a product of vectors and matrices, and an efficient computational scheme for generating these polynomials.