Equivalence of a Scalar and a Vector Equilibrium Problem for a Pair of Functions Forming a Nikishin System

Abstract

We prove the equivalence of the vector and scalar equilibrium problems which arise naturally in the study of the limit zeros distribution of type I Hermite-Padé polynomials for a pair of functions forming a Nikishin system. Bibliography: 22 titles.

Scalar equilibrium problem and the limit distribution of the zeros of Hermite-Padé polynomials of type II

“…1.1. The present paper continues the studies initiated by the second author in [28] and [33]. In these papers, a new scalar approach to the problem on the limit distribution of the zeros of Hermite-Padé polynomials for a pair of functions forming a Nikishin system was proposed and shown to be equivalent to the traditional vector approach (see [33], Theorem 1).…”

“…The present paper continues the studies initiated by the second author in [28] and [33]. In these papers, a new scalar approach to the problem on the limit distribution of the zeros of Hermite-Padé polynomials for a pair of functions forming a Nikishin system was proposed and shown to be equivalent to the traditional vector approach (see [33], Theorem 1). We recall that the traditional approach to this problem is based on the solution of a vector equilibrium problem in potential theory with a 2 × 2-matrix (known as the Nikishin matrix); see, first of all, [21], [22], [9], and [10], and also [1], [4], [17], and [18], and the references given therein.…”

“…In [28] it was shown that the limit distribution of the zeros of Hermite-Padé polynomials of type I coincides with the measure λ = π 2 (λ), where π 2 is the canonical projection (a two-sheeted covering) of the Riemann surface under consideration to the Riemann sphere. In the present paper, the scalar equilibrium measure λ is used to characterize the limit distribution of the zeros of Hermite-Padé polynomials of type II for the same pair of functions f 1 and f 2 as in the papers [28] and [33] (see Section 1.2 below). As was already pointed out, on an example of two Markov functions f 1 and f 2 considered here (see formula (1.1) below) it was shown in [33] that these two approaches (the vector and scalar ones) are equivalent.…”

Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces

“…Thus, in the present paper, we further extend the new approach (which was proposed by the author in [30]) to the study of asymptotic properties of Hermite-Padé polynomials for multivalued functions. This approach is based on the extremal equilibrium problem posed not on the Riemann sphere, but instead on the Riemann surface R 2 (w) (further advances in this problem were made in [33], [12]).…”

“…(26) The facts that in (26) the identity holds on the entire compact set K and supp λ K = K were proved in [33].…”