Hermite-Padé approximants for meromorphic functions on a compact Riemann surface

Abstract

For an arbitrary tuple of m + 1 germs of analytic functions at a fixed point, we introduce the so-called polynomial Hermite-Padé m-system (of order n, n ∈ N), which consists of m tuples of polynomials; these tuples, which are indexed by a natural number k ∈ [1, . . . , m], are called the kth polynomials of the Hermite-Padé m-system. We study the weak asymptotics of the polynomials of the Hermite-Padé m-system constructed at the point ∞ from the tuple of germs [1, f 1,∞ , . . . , f m,∞ ] of the functions 1, f 1 , . . . , f m that are meromorphic on some (m + 1)-sheeted branched covering π : R → C of the Riemann sphere C of a compact Riemann surface R. In particular, under some additional condition on π, we find the limit distribution of the zeros and the asymptotics of the ratios of the kth polynomials for all k ∈ [1, . . . , m]. It turns out that in the case, where f j = f j for some meromorphic function f on R, the ratios of some kth polynomials of such Hermite-Padé m-system converge to the sum of the values of the function f on the first k sheets of the Nuttall partition of the Riemann surface R into sheets. Contents 1 Introduction 2 Statements of the main results 3 Reconstruction of the values of a function meromorphic on R from its germ via the polynomial Hermite-Padé m-system 4 The Riemann surface R [k] and the definition of the kth polynomials of the Hermite-Padé m-system in terms of this surface 5 Proofs of Theorems 1 and 2 6 The condition of connectedness of the Riemann surface R [k]

Метод Задачи Римана-Гильберта В Применении К Системе Никишина

“…The Riemann-Hilbert analysis for ray sequences of indices, where n/m → γ, was recently done in [38] (for an Angelesco system) and [3] (for Frobenius-Padé approximants). There is also high interest in the asymptotics of type I Nikishin systems with complex singular points, see [34,22,23] and the references therein.…”

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

“…At present, the answer to the problem of the limit distribution of the zeros of Hermite-Padé polynomials is available only for some particular classes of analytic functions (see [17], [34], [38], [19], [2], [4], [40], [32]). As a rule, the limit distribution of the zeros of Hermite-Padé polynomials for a pair of functions f 1 , f 2 can be described following the approach first proposed by Nuttall (see [36], [38]) in terms related to some three-sheeted Riemann surface which in a certain sense 1 is "associated" with the pair of functions f 1 , f 2 (for the relation between the three-sheeted Riemann surface with the asymptotics of Hermite-Padé polynomials, see also [25], [6], [26].) For a pair of functions f 1 , f 2 of form (1) the above problem was solved by Nikishin [34] in 1986 (see also [33], [35], [7]).…”

Maximum Principle and Asymptotic Properties of Hermite--Padé Polynomials

“…Combining (18), ( 26) and (30), we obtain finally that for any admissible ρ 2 , ρ ∈ (1, R) and as n → ∞ the following asymptotic relations valid…”

“…Then f is a (single-valued) meromorphic function on the RS R 4 (w). Therefore the asymptotic properties of type I Hermite-Padé polynomials (HP-polynomials) for the collection of four functions [1, f, f 2 , f 3 ] follows directly from the results of the paper [18] (see also [19] and [20]). But this is not the case for the collection of three functions [1, f, f 2 ] when f ∈ C(z, w).…”