Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 374.30030
Autor: Erdös, Paul; Reddy, A.R.
Title: Addendum to "Rational approximation". (In English)
Source: Adv. Math. 25, 91-93 (1977).
Review: Let f(z) = sumk = 0ooakzk,a0 > 0,ak \geq 0(k \geq 1) be an entire function such that l < lim\supr ––> oo\frac{log log M(r)}{log log r} = \rho+1 < oo and lim\supr ––> oo(inf) (log r)-\rho-1 log M(r) = \alpha(\beta) where M(r) = max||z|| = r||f(z)||. The authors in Adv. Math. 21, 78-109 (1976; Zbl 334.30019) showed that if (*) 5 < 2\beta < 2\alpha < oo then for every sequence of polynomials {Pn(x)}n = 0oo of degree at most n, liminfn ––> oo\deltann^{-1-\rho^{-1}} \geq e-1 where \deltan = ||\frac1{f(x)}-\frac 1{Pn(x)}||L_{oo[0,oo]}.
In the present paper, they show that if (*) is replaced by (**) 0 < \beta \leq \alpha < oo, then for every polynomial Pn(x) and Qn(x) of degree at most n, liminfn ––> oo\trianglenn^{-1-\rho^{-1}} \geq G where \trianglen = ||\frac 1{f(x)}-\frac{Pn(x)}{Qn(x)}||L\_{oo[0,oo]} and G = \exp\left{-(\frac 2{\beta})1/\rho[\alpha -1+(\frac{2\alpha}{\beta})1/(\rho+1)]\right}.
Reviewer: O.P.Juneja
Classif.: * 30E10 Approximation in the complex domain
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