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Biorthogonal Polynomial Bases and Vandermonde‐like Matrices
Author(s) -
VerdeStar Luis
Publication year - 1995
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1995953269
Subject(s) - vandermonde matrix , mathematics , polynomial , chebyshev nodes , biorthogonal system , computation , matrix (chemical analysis) , alternating polynomial , chebyshev polynomials , companion matrix , algebra over a field , pure mathematics , matrix polynomial , polynomial matrix , mathematical analysis , eigenvalues and eigenvectors , algorithm , physics , materials science , wavelet transform , quantum mechanics , artificial intelligence , computer science , wavelet , composite material
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 3 N 2 .