Biorthogonal Polynomial Bases and Vandermonde‐like Matrices
Luis Verde‐Star · Studies in Applied Mathematics · 1995
This article considers a family of Gram matrices of pairs of bases of a finite dimensional vector space of polynomials with respect to certain indefinite inner products. Such a family includes all the generalized confluent Vandermonde matrices relative to any polynomial basis, like the Chebyshev‐Vandermonde matrices, for example. Using the biorthogonality of pairs of bases with respect to a divided difference functional, properties of matrices and functionals, as well as interpolation formulas are obtained. I show that the computation of the inverse of a Vandermonde‐like matrix is essentially equivalent to the computation of the partial fractions decompositions of a set of rational functions with a common denominator. I also explain why the various Chebyshev‐Vandermonde matrices are the simplest generalizations of the classic Vandermonde matrices and describe a simple algorithm for the computation of their inverses, which requires a number of multiplications of the order of 3N2.