Asymptotic Form of Hermite-Padé Polynomials

Asymptotics of Hermite-Padé polynomials for a set of functions with different branch points

“…Now, in this case, o)(z) has an analytic continuation away from Fj, and the particular choice of location of F~ is immaterial in determining the behavior of fj(z) near z =oo and thus the form of pj(z),j= 1, 2, 3. However, as in previous problems of this sort [1], we find that a particular choice of Fj,j = 2, 3, is required to most conveniently specify and obtain the asymptotic behavior of {pj-(z)} as n ~ c~. Only if the points bl,..., b4 satisfy certain conditions will a choice of {Fj} exist for which the results of this paper hold.…”

“…In the construction of a reproducing kernel we shall use polynomials p~~ of degree n + 1 with zeros at the n + 1 zeros of gjt)(z) near Fj and the same behavior at ~ as g~~ (1). 4 Collecting these results, and taking the limits R -~ oo, p -, 0, using (3.8), we find Since p~~ X~~ are analytic, nonzero in a neighborhood of Fj, the argument of the exponential must be analytic there.…”

Asymptotics of Hermite-Pade Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0)

“…we obtain the last part of the theorem [8]; the equivalent form of the algebraic equation for the Riemann surface of the case III goes back to Nuttall [62,12]). The proofs of Theorem 2.18 and Proposition 2.17 are a repetition of the proofs of the corresponding results for the case I in Theorem 2.10 and Proposition 2.9.…”

“…The analytic theory of Hermite-Padé approximants for the complex case has been initiated by Nuttall. In the two pioneering papers [12,61] of 1981 he obtained some asymptotic formulas for Hermite-Padé approximants to functions with separated complex branch points [12] (a complex analog of an Angelesco system) and to functions meromorphic on the same Riemann surface [61] (i.e., functions with the same set of branch points, like a Nikishin system for the real case). The results of [12] were verified by some heuristic considerations and numerical experiments, and the paper [61] contains rigorous theorems.…”

Аппроксимации Эрмита - Паде И Ансамбли Совместно Ортогональных Многочленов

“…В двух пионерских работах [38], [39] 1981 г. он получил несколько асимпто-А. И. АПТЕКАРЕВ, А. Б. Э. КОЙЭЛААРС тических формул для аппроксимаций Эрмита-Паде функций с разделенными комплексными точками ветвления [38] (комплексный аналог системы Анжеле-ско) и для функций, мероморфных на одной и той же римановой поверхно-сти [39] (т. е. имеющих одно и то же множество точек ветвления, аналогично системе Никишина в вещественном случае). В [38] результаты обосновывались эвристическими рассуждениями и вычислительным экспериментом, в то вре-мя как [39] содержит строго доказанные теоремы.…”