Padé tables of entire functions of very slow and smooth growth

Abstract

Let f(z) = E~~ be entire with ]aj_la~+l/a~] _< p2o, j = 1.2,3 ..... where Po = 0.4559... is the positive root of the equation 2 ~ pJ~ = 1. j=l It is shown that the Pad6 table of f is normal, and as L --* oo, [L/ML](z ) converges uniformly in compact subsets of C to f, for any sequence of nonnegative integers { ML }~= x. In particular, the diagonal sequence {[ L/L]} converges uniformly in compact subsets of C to f. Furthermore, the constant O0 is shown to be best possible in a strong sense.

Тригонометрические Аппроксимации Паде Функций С Регулярно Убывающими Коэффициентами Фурье

“…В алгеб-раическом случае аналогичные задачи изучались А. Левиным и Д. Любински (см. [11]- [14]) в связи с анализом результатов Э. Саффа (см. [15], [16]), относя-щихся к рациональной аппроксимации экспоненты.…”

“…[15], [16]), относя-щихся к рациональной аппроксимации экспоненты. В работах [11]- [14], [17]- [19] найдены широкие классы целых функций, для которых классические аппрокси-мации Паде π n,m ( · ; f ) приближают f в единичном круге со скоростью, асимп-тотически равной наилучшей, т.е. наблюдается так называемый эффект Саф-фа (подробнее см.…”

Padé approximants to certain elliptic-type functions

“…For restricted classes of functions, such approximants were proven to converge, locally uniformly in the domain of analyticity, as n goes large. These classes include Markov functions and rational perturbation thereof [37,23,47,12], Cauchy transforms of continuous non-vanishing functions on a segment [11,42,36] (in these examples interpolation takes place at infinity), and certain entire functions such as Polya frequencies or functions with smooth and fast decaying Taylor coefficients [5,33,34] (interpolation being now at the origin). However, such favorable cases do not reflect the general situation which is that Padé approximants often fail to converge locally uniformly, due to the occurrence of "spurious" poles that may wander about the domain of analyticity.…”

Best Rational Approximations of Entire Functions Whose Maclaurin Series Coefficients Decrease Rapidly and Smoothly

“…It was shown in [3] that the Padé table of / is normal, and that any sequence of Padé approximants with numerator degrees tending to infinity converges to / uniformly on compact subsets of C. These properties of the Padé approximants (together with their additional properties to be established in §2) enable us to use Saff s argument in a modified form to prove results analogous to Theorems A, B and C for this class of functions.…”

“…Recently one of the authors [3,4] investigated the Padé tables of entire functions such as 00 (1) f(z)= ZajzJ, fly* O,;'= 0,1,2,..., 7 = 0 which satisfy (2) \aj_xaJ + x/a2\^p2, j =1,2,..., where (3) P < p0, and p0 = 0.4559... is the positive root of the equation 2E"_1pj'" = 1.…”