BIORTHOGONAL POLYNOMIAL BASES AND VANDERMONDE-LIKE MATRICES

This article considers a family of Gram matrices of pairs of bases of a finite dimensional vector space of polynomials with respect to certa in indefinite inner products, Such a family includes all the generaliz ed confluent Vandermonde matrices relative to any polynomial basis, li ke the Chebyshev-Vandermonde matrices, for example. Using the biorthog onality of pairs of bases with respect to a divided difference functio nal, 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-V andermonde matrices are the simplest generalizations of the classic Va ndermonde matrices and describe a simple algorithm for the computation of their inverses, which requires a number of multiplications of the order of 3N(2).