21 research outputs found
Szeg\H{o}-type asymptotics for ray sequences of Frobenius-Pad\'e approximants
Let be a Cauchy transform of a possibly complex-valued Borel
measure and be a system of orthonormal polynomials with
respect to a measure ,
. An -th
Frobenius-Pad\'e approximant to is a rational function ,
, , such that the first
Fourier coefficients of the linear form vanish when the
form is developed into a series with respect to the polynomials . We
investigate the convergence of the Frobenius-Pad\'e approximants to
along ray sequences , , when
and are supported on intervals on the real line and their
Radon-Nikodym derivatives with respect to the arcsine distribution of the
respective interval are holomorphic functions
Discrete integrable systems generated by Hermite-Pad\'e approximants
We consider Hermite-Pad\'e approximants in the framework of discrete
integrable systems defined on the lattice . We show that the
concept of multiple orthogonality is intimately related to the Lax
representations for the entries of the nearest neighbor recurrence relations
and it thus gives rise to a discrete integrable system. We show that the
converse statement is also true. More precisely, given the discrete integrable
system in question there exists a perfect system of two functions, i.e., a
system for which the entire table of Hermite-Pad\'e approximants exists. In
addition, we give a few algorithms to find solutions of the discrete system.Comment: 20 page
Multidimensional Toda Lattices: Continuous and Discrete Time
In this paper we present multidimensional analogues of both the continuous-
and discrete-time Toda lattices. The integrable systems that we consider here
have two or more space coordinates. To construct the systems, we generalize the
orthogonal polynomial approach for the continuous and discrete Toda lattices to
the case of multiple orthogonal polynomials
Convergence of ray sequences of Frobenius-Padé approximants
Let be a Cauchy transform of a possibly complex-valued Borel measure and a system of orthonormal polynomials with respect to a measure , where . An th Frobenius-Padé approximant to is a rational function , , , such that the first Fourier coefficients of the remainder function vanish when the form is developed into a series with respect to the polynomials . We investigate the convergence of the Frobenius-Padé approximants to along ray sequences , , when and are supported on intervals of the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the corresponding interval are holomorphic functions
On 2D discrete Schr\"odinger operators associated with multiple orthogonal polynomials
A class of cross-shaped difference operators on a two dimensional lattice is
introduced. The main feature of the operators in this class is that their
formal eigenvectors consist of multiple orthogonal polynomials. In other words,
this scheme generalizes the classical connection between Jacobi matrices and
orthogonal polynomials to the case of operators on lattices. Furthermore we
also show how to obtain 2D discrete Schr\"odinger operators out of this
construction and give a number of explicit examples based on known families of
multiple orthogonal polynomials.Comment: 15 page
On the parametrization of a certain algebraic curve of genus 2
A parametrization of a certain algebraic curve of genus 2, given by a cubic equa-tion, is obtained. This curve appears in the study of Hermite-Pade´ approximants for a pair of functions with overlapping branch points on the real line. The suggested method of parametrization can be applied to other cubic curves as well