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

Abstract

A new approach to the problem of the zero distribution of Hermite-Padé polynomials of type I for a pair of functions f1, f2 forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field, which is posed on a two-sheeted Riemann surface.Bibliography: [50] titles. Paper: http://mi.mathnet.ru/eng/tm3908

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]).…”

“…According to [31] (see also [19] and [13]), this approach in principle allows one to double (in comparison with the Stahl domain) the domain in which the values of the function f can be efficiently (that is, in terms of rational functions) recovered from a given germ. The corresponding generalizations within the framework of this new approach are given in [31] also for the family [1, f, f 2 , f 3 ] (see also [30], [34]). In this case, the domain of effective recovery of an analytic function increases already by three times in comparison with the Stahl domain.…”

“…In the last § 4, we compare the extremal Problem 1 considered here and the extremal problem of [30] on the existence of a three-sheeted Nuttall-partitioned Riemann surface associated with a given multivalued analytic function.…”

“…Nevertheless, it turns out the so-called strong asymptotics of the Padé polynomials is characterized precisely in terms pertaining to this two-sheeted Riemann surface R 2 (f ∞ ) (see [19], [28], [3]). So, our approach, in which properties of extremal compact sets pertaining to Hermite-Padé polynomials are studied with the help of the results obtained earlier in the Stahl theory and its further advances made by Stahl himself and other researchers (see [28], [30], [32], [12]), is also quite natural. The new approach proposed in [30] has proved instrumental in delivering, for the class of functions of the form (2), some new and previously available results related to the Hermite-Padé polynomials in terms of the scalar equilibrium problem (posed on a two-sheeted Riemann surface), rather than in terms of the generally accepted equilibrium problem (posed on the Riemann sphere).…”

Tschebyshev-Padé approximations for multivalued functions

“…The first on is related to the construction of the proper generalization of the notion of S-compact set. The idea of how to solve this problem was proposed in the papers [43], [40], [41], [44], [21]. The new approach to the existence problem developed in these papers is based on consideration of an equilibrium problem on a double-sheeted Riemann surface.…”