Распределение Нулей Полиномов Паде И Аналитическое Продолжение

Abstract

Обсуждается задача аналитического продолжения многозначной аналитической функции с конечным множеством точек ветвления на римановой сфере. Основное внимание уделяется аппроксимациям Паде-классическим (одноточечным), многоточечным аппроксимациям Паде и аппроксимациям Эрмита-Паде. Основной результат работы-теорема о распределении нулей и сходимости аппроксимаций Эрмита-Паде для набора [1, f, f 2 ], где многозначная функция f принадлежит так называемому классу Лагерра L. Библиография: 128 названий. Ключевые слова: аналитическое продолжение, непрерывные дроби, ортогональные многочлены, рациональные аппроксимации, полиномы Паде, полиномы Эрмита-Паде, распределение нулей, GRS-метод, сходимость по емкости.

Zero distribution for Angelesco Hermite–Padé polynomials

“…The conjecture is supported by the results from [44] related to function f from the class f k ∈ L of the form f k (z) = m k i=1 (z − a i,k ) α i,k with m k i=1 α i,k = 0 (see also [33] and [68]). For function of this class the weighted Hermite-Padé polynomials q n,k (z)f k (z) and the remainder satisfy the same differential equation with polynomial coefficient of order s + 1 (polynomials -coefficient depend on n but their degrees are bounded by constants not depending on n).…”

“…In the connection to the Markov case see also some more recent results in [7] and [54] where mixed case has been studied for two Markov functions in case when one support is a subinterval of another one. Nikishin case is considered in [38], [9]; see also [56], [68], [70].…”

“…We note that in the case of multipoint Padé approximants the complement to the extremal compact set is not necessary a domain, i.e., it might consists of several domains; see [16], [17], [18]. In the case of Hermite-Padé polynomials for the general complex Nikishin systems the situation is very similar to the multipoint Padé approximants; see [43], [56], [68].…”

“…which has been used in sec 4.2.5 for the case s = 1. The closure Ω k is defined on the basis of the inner metric in Ω k same way as in ( 67) and (68) where in place of Γ k we use Γ 1 k for positive k and Γ for k = 0. Then, again like in case s = 1, we introduce the equivalency relation ∼ in the topological sum Σ = s k=0 Ω k joining in a standard way the copy of (Γ 1 k ) + on the sheet zero with the copy of (Γ 1 k ) − on the k-the sheet and vise versa.…”

Strong asymptotics of Hermite-Pade approximants for the Nikishin system of Jacobi weights

“…Приложениям в теории аппроксимации посвяще-ны работы [46], [51]. Приложениям в теории вероятностей -работы [6], [31], [32].…”