Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s (d + 1 ) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials. We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions. We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite-Padé approximation in matrix form is given.

[1]  Adhemar Bultheel,et al.  A general module theoretic framework for vector M-Padé and matrix rational interpolation , 2005, Numerical Algorithms.

[2]  J. Arvesú,et al.  Some discrete multiple orthogonal polynomials , 2003 .

[3]  P. Maroni.  L'orthogonalité et les récurrences de polynômes d'ordre supérieur à deux , 1989 .

[4]  Recursive relations for block Hankel and Toeplitz systems part II: Dual recursions , 1984 .

[5]  Erik Koelink,et al.  Review of ``Classical and quantum orthogonal polynomials in one variable. "by Mourad E.H. Ismail , 2011 .

[6]  G. L. Lagomasino,et al.  HERMITE-PADÉ Approximation for Nikishin Systems of Analytic Functions , 1994 .

[7]  P. Maroni,et al.  Two-dimensional orthogonal polynomials, their associated sets and the co-recursive sets , 1992, Numerical Algorithms.

[8]  V. N. Sorokin,et al.  Rational Approximations and Orthogonality , 1991 .

[9]  Bernhard Beckermann,et al.  A reliable method for computing M-Pade´ approximants on arbitrary staircases , 1992 .

[10]  A. Aptekarev,et al.  Multiple orthogonal polynomials , 1998 .

[11]  Adhemar Bultheel,et al.  Recursive algorithms for the matrix Padé problem , 1980 .

[12]  M. Ismail,et al.  Classical and Quantum Orthogonal Polynomials in One Variable: Bibliography , 2005 .

[13]  George Labahn,et al.  A uniform approach for Hermite Padé and simultaneous Padé approximants and their matrix-type generalizations , 1992, Numerical Algorithms.

[14]  B. Beckermann,et al.  On the Definition and Normality of a General Table of Simultaneous Padé Approximants , 1994 .

[15]  Francisco Marcellán,et al.  On recurrence relations for Sobolev orthogonal polynomials , 1995 .

[16]  J. V. Iseghem.  Vector orthogonal relations. Vector QD-algorithm , 1987 .

[17]  V. Kaliaguine.  The operator moment problem, vector continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials , 1995 .

[18]  D. W. Lee,et al.  Difference equations for discrete classical multiple orthogonal polynomials , 2008, J. Approx. Theory.

[19]  A. J. Durán.  A Generalization of Favard's Theorem for Polynomials Satisfying a Recurrence Relation , 1993 .

[20]  P. Maroni,et al.  A characterization of “classical” d -orthogonal polynomials , 1995 .

[21]  Walter Van Assche,et al.  Some classical multiple orthogonal polynomials , 2001 .

[22]  A. F. Moreno,et al.  Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials , 2010 .

[23]  Jeannette Van Iseghem,et al.  Algebraic Aspects of Matrix Orthogonality for Vector Polynomials , 1997 .

[24]  W. Assche,et al.  Differential equations for multiple orthogonal polynomials with respect to classical weights: raising and lowering operators , 2006 .

[25]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[26]  W. Assche.  Analytic number theory and approximation , 2008 .

[27]  W. Van Assche,et al.  Multiple orthogonal polynomials for classical weights , 2003 .