Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  272.41007
Autor:  Erdös, Paul; Reddy, A.R.
Title:  Rational approximation to certain entire functions in [0,+oo). (In English)
Source:  Bull. Am. Math. Soc. 79, 992-993 (1973).
Review:  Let f(z) = sum ook = 0akzk, a0 > 0 and ak \geq 0 (k \geq 1) be an entire function. Denote

\lambda0,n = \lambda0,n(1/f) = infpn in \pin |{1 \over f(x)}-{1 \over pn(x)} |Loo [0, oo)

where \pin denotes the clan of all polynomials of degree at most n. By using the above notation the authors announce the following. (i) For each \epsilon > 0, there exists infinitely many m for which \lambda0,m \leq \exp ({-m \over (log m)1+\epsilon} ). ii) For each \epsilon > 0 there exist a subsequence of natural numbers for which \lambda0,m \geq \exp (- \epsilon m). (iii) Let aj = qjk, where 0 < q < 1, 2 \leq k < oo. Then

q = liminf(\lambda0,n)n^{1/k} \leq limsup(\lambda0,n)n^{k/1} \leq q1-2^{1-k}.

Classif.:  * 41A20 Approximation by rational functions
                   41A50 Best approximation
                   41A25 Degree of approximation, etc.

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