Scalar Equilibrium Problem and the Limit Distribution of Zeros of Hermite-Padé Polynomials of Type II

Abstract

The main purpose of the paper is to present some powerful data on the advantage of the rational approximation procedure based on Hermite-Padé polynomials over the Padé approximation procedure. The first part of the paper is devoted to some numerical examples in this direction. The second part will be devoted to some theoretical results.In particular, we demonstrate our ideas about the advantage of rational Hermite-Padé approximants over Padé approximants analyzing the analytical structure of the frequency function ν of the free Van der Pol equation.Bibliography: [41] titles.

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]).…”

“…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).…”

“…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). This scalar approach was further advanced in [29], [30], [32], [12] for a pair of functions forming a Nikishin system (cf. [21]).…”

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.…”