Christoffel transformation for a matrix of bi-variate measures. (English) Zbl 1437.42038
Bi-orthogonal polynomial systems are in the meantime well-known generalisations of classical orthogonal polynomials from numerical analysis and approximation theory (used, e.g., for Gauß-quadrature formulae). They are applied in this paper in matrix form, so even more general than the already large class of biorthogonal orthogonal polynomials. The authors consider in particular bi-orthogonal matrix orthogonal Laurent polynomials and study pertubations of (for instance) positive definite linear functionals on infinite subsets of \(\mathbb R\) (cf. functionals studied by E. B. Christoffel [J. Reine Angew. Math. 55, 61–82 (1858; ERAM 055.1450cj)] for Christoffel matrix transformations, J. Geronimus [Izv. Akad. Nauk SSSR, Ser. Mat. 4, 215–228 (1940; Zbl 0025.04104)] and V. B. Uvarov [Dokl. Akad. Nauk SSSR 126, 33–36 (1959; Zbl 0087.06102); U.S.S.R. Comput. Math. Math. Phys. 9, No. 6, 25–36 (1972; Zbl 0231.42013)]). The pertubations and the original linear functionals are to be compared with each other, for which several methods are available, mostly using a variety of matrix factorisations. The theory becomes more complicated if the infinite subsets of the real line are replaced by unit circles. In this article in particular, Christoffel matrix transformations on double circles on compactly supported measures are analysed in much detail.
Reviewer: Martin D. Buhmann (Gießen)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 15A23 | Factorization of matrices |
| 30C10 | Polynomials and rational functions of one complex variable |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
matrix biorthogonal polynomials; matrix-valued measures; nondegenerate continuous bilinear forms; Gauss-Borel factorization; matrix Christoffel transformations; quasideterminants; block CMV matricesReferences:
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