35 research outputs found
Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials
Spectral and factorization properties of oscillatory matrices lead to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue–Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is givenpublishe
Generating new classes of orthogonal polynomials
Given a sequence of monic orthogonal polynomials (MOPS), {Pn}, with respect
to a quasi-definite linear functional u, we find necessary and sufficient conditions on the parameters
an and bn
for the sequence
Pn(x)+anPn−1(x)+bnPn−2(x),   n≥1P0(x)=1,P−1(x)=0
to be orthogonal. In particular, we can find explicitly the linear functional v such that the new
sequence is the corresponding family of orthogonal polynomials. Some applications for Hermite
and Tchebychev orthogonal polynomials of second kind are obtained
Quadratic decomposition of bivariate orthogonal polynomials
We describe the relation between the systems of bivariate
orthogonal polynomial associated to a symmetric weight function and
associated to some particular Christoffel modifications of the quadratic
decomposition of the original weight. We analyze the construction of a
symmetric bivariate orthogonal polynomial sequence from a given one,
orthogonal to a weight function defined on the first quadrant of the
plane. In this description, a sort of Backlund type matrix transformations
for the involved three term matrix coefficients plays an important
role. Finally, we take as a case study relations between the classical
orthogonal polynomials defined on the ball and those on the simplex.publishe
Riemann–Hilbert problem and matrix biorthogonal polynomials
Recently the Riemann-Hilbert problem, with jumps supported on appropriate
curves in the complex plane, has been presented for matrix biorthogonal polynomials, in
particular non-Abelian Hermite matrix biorthogonal polynomials in the real line, understood
as those whose matrix of weights is a solution of a Sylvester type Pearson equation
with coe cients first order matrix polynomials. We will explore this discussion, present
some achievements and consider some new examples of weights for matrix biorthogonal
polynomials.publishe
Matrix Jacobi Biorthogonal Polynomials via Riemann-Hilbert problem
We consider matrix orthogonal polynomials related to Jacobi type matrices of
weights that can be defined in terms of a given matrix Pearson equation.
Stating a Riemann-Hilbert problem we can derive first and second order
differential relations that these matrix orthogonal polynomials and the second
kind functions associated to them verify. For the corresponding matrix
recurrence coefficients, non-Abelian extensions of a family of discrete
Painlev\'e d-PIV equations are obtained for the three term recurrence relation
coefficients
Matrix Toda and Volterra lattices
We consider matrix Toda and Volterra lattice equations and their relation with matrix biorthogonal polynomials. From that relation, we give a method for constructing a new solution of these systems from another given one. An illustrative example is presented.publishe