Matrix biorthogonal polynomials on the real line: Geronimus transformations Articles uri icon

publication date

  • August 2018

start page

  • 1

end page

  • 66

international standard serial number (ISSN)

  • 1664-3607

electronic international standard serial number (EISSN)

  • 1664-3615

abstract

  • In this paper Geronimus transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasidefinite matrix of bivariate generalized functions with a well defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The results are mainly concerned with the derivation of Christoffel type formulas, which allow to express the perturbed matrix biorthogonal polynomials and its norms in terms of the original ones. The basic tool is the Gauss&-Borel factorization of the Gram matrix, and particular attention is paid to the non-associative character, in general, of the product of semi-infinite matrices. The Geronimus transformation, in where a right multiplication by the inverse of a matrix polynomial and an addition of adequate masses is performed, is considered. The resolvent matrix and connection formulas are given. Two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. This approach requires the perturbing matrix polynomial to have a nonsingular leading term. Then, using spectral techniques and spectral jets, Christoffel&-Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques, that is valid also for singular leading coefficients. When the leading term is nonsingular a comparative of between both methods is presented. ...

keywords

  • Matrix biorthogonal polynomials
    Spectral theory of matrix polynomials
    Quasidefinite matrix of generalized kernels
    Nondegenerate continuous sesquilinear forms
    Gauss–Borel factorization
    Matrix Geronimus transformations
    Matrix linear spectral transformations
    Christoffel type formulas
    Quasideterminants
    Spectral jets
    Unimodular matrix polynomials