Hermite‐Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports

Abstract

Abstract. Hermite-Padé approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szegő-type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann-Hilbert problem for multiple orthogonal polynomials (the common denominator).

Differential equations for the recurrence coefficients limits for multiple orthogonal polynomials from a Nevai class

“…3) We also can affirm (at least for d = 2) that Theorem 2 remains valid for the case of touching intervals (technicalities can be taken from [7]) and for weight functions (1.17) with singularities of the types: Jacobi and Fisher-Hartwig weights [18].…”

Riemann-Hilbert analysis for a Nikishin system

“…If the intervals [a, b] and [c, d] are touching, then the construction of the parametrix around the common point a = d also requires a local 3 × 3 Riemann-Hilbert problem, but it is not clear what such a parametrix should contain. We believe it will be somewhat like the local parametrix which was used in [14] around the common point of the two intervals in an Angelesco system, or the parametrix used in [7] for the critical case a = 1/ √ 2 in that paper, but it will not be quite the same parametrix because in an Angelesco system the two intervals are repelling, whereas in a Nikishin system the two intervals are attracting. Also the critical case c = c * is not considered in this paper.…”

On the Parametrization of a Certain Algebraic Curve of Genus 2

“…Здесь при определении неизвестных коэффициентов у ( ) = + · · · , = 1, 2, используется условие мнимости периодов абелевых интегралов Re ∮︁ = 0, где интегрирование ведется вдоль любого цикла римановой поверхности R алгебраической функции (1), (4). Для случаев рода 1 и 2 у кривых (1), (4) в [3] предложены эллиптические и соответственно ультраэллиптические униформизации. В случае рода 0 в [4] найдены рациональные униформизации.…”