Riemann-Hilbert analysis for a Nikishin system

Abstract

In this paper we give the asymptotic behavior of type I multiple orthogonal polynomials for a Nikishin system of order two with two disjoint intervals. We use the Riemann-Hilbert problem for multiple orthogonal polynomials and the steepest descent analysis for oscillatory Riemann-Hilbert problems to obtain the asymptotic behavior in all relevant regions of the complex plane.

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

“…Remark 2. It is well known (see [1], and also [4] and [29]) that, for a pair of functions f 1 , f 2 forming a Nikishin system, the support of the equilibrium measure λ in the diagonal case (which is considered here) coincides with the entire compact set F ; this means that only the case of identical equality in relations (16) is possible. So far, this fact has not yet been proved within the framework of the approach proposed here.…”

“…forming an Angelesco system. The author is grateful to the referee for the many helpful comments and suggestions which led to a great improvement in the presentation of the paper and for calling his attention to the papers [1] and [29].…”

Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials

“…The so called Nikishin systems of measures introduced in [18] play a central role in many of these studies. Some of the basic questions involve uniqueness of the MOP [7], convergence of the corresponding Hermite-Padé approximants [4], nth-root [8], ratio [3] (see also [13]), and strong [1,16] asymptotics of sequences of MOP. We have limited to a short list of significant contributions, see also reference lists in [14,15].…”

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

“…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. The vector equilibrium problem is known to have a unique solution given by vector measure λ = (λ 1 , λ 2 ).…”

“…from (1.1), (1.2), and (1.3) it follows that ∆f 2 (x)/∆f 1 (x) = σ(x), x ∈ (−1, 1), where ∆f j (x) is the difference of the limit values (the jump) of the function f j , j = 1, 2, in the upper and lower half-planes, respectively. It follows that the pair of functions (f 1 , f 2 ) forms a Nikishin system (for more on such systems, see [21], [22], and also [2], [4], [18], and the references given therein). Note that in the paper [31] an example of a multivalued analytic function f is given such that the pair of functions f, f 2 forms a Nikishin system (under a minimal extension of the definition of a Nikishin system compared to the classical one).…”