Hermite-Padé approximation of exponential functions

Abstract

В работе изучаются свойства диагональных многочленов Эрмита-Паде 1-го рода для системы экспонент {e λ j z } k j=0 с произвольными различными комплексными параметрами {λ k } k j=0 : установлена асимптотика остаточной функции, описана область локализации нулей; при действительных значениях параметров найдены асимптотики и описаны экстремальные свойства. Доказанные теоремы дополняют известные результаты П. Борвейна, Ф. Вилонского, Э. Саффа и Р. Варги, Г. Шталя. Библиография: 43 названия.

Development of a method for forecasting random events during instability periods

“…In accordance with the main theorem of higher algebra [17], a polynomial of arbitrary degree (say, N degree) with real coefficients has exactly N roots that are either real or create complex conjugate pairs. Then the denominator of polynomial (2) and the purely formal denominator of polyno mial (9) have exactly M roots.…”

“…which, in essence, plays the role of analytic elongation of polynomial (2) onto the plane of the complex variable z, will be stable. A positive answer to this question will take place when the convergence condition for the Pad approximation in the unit circle of the zplane is satisfied [17]. This condition is satisfied relatively simply.…”

“…Thus, when applying the Pad approximation, it is necessary to control the absolute values of the poles of the approximating polynomial. This is a kind of fee that it is possible to pay for the high accuracy of the Pad approximations and their convergence to horizontal asymptotes even with a low order of the approximating polyno mial [17,19]. In this regard, let's note that the approximating polynomial will retain its properties, the desire for horizontal asymptotes (in particu lar, the abscissa axis) when a simple condition is met.…”

“…Thus, the results of the analysis allow to conclude that approximation of a number of finely rational functions or the socalled Pad approximation can give good results [16,17]. Such an approach should take advantage of polynomial approximation.…”