Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  347.41009
Autor:  Erdös, Paul; Reddy, A.R.
Title:  Rational approximation on the positive real axis. (In English)
Source:  Proc. Lond. Math. Soc., III. Ser. 31, 439-456 (1975); corrigendum ibid. 35, 290 (1977).
Review:  This paper is a continuation of the authors' researches on the problem of approximating the reciprocal of an entire function with nonnegative Taylor coefficients in the uniform norm on the positive half-axis by means of reciprocals of polynomials. [previous papers Bull. Amer. math. Soc. 79, 992-993 (1973; Zbl 272.41007) and Period. math. Hungar 6, 241-244 (1975; Zbl 273.41012), ibid. 7, 27-35 (1976; Zbl 337.41020)]. Quoting the authors this paper may serve as a guide to those interested in this topic. From the results we quote: Letting \lambda0,n denote the degree of approximation with nth degree polynomials in the problem described above, we have: For any \epsilon > 0 and k \geq 1 there exist infinitely many n such that

\lambda0,n \leq \exp (-n/ log n log log n ... (log (k)n)1+\epsilon)

but there exists to every k and every large c a function such that

\lambda0,n \geq \exp (-cn/ log n log log n log (k)n).

For an entire function of order \rho, 0 < \rho < oo, type \tau and lower type \omega we have

limsup \lambda \rho/n0,n \leq \exp (- \omega /(e+1) \tau).

The paper also contains lower estimates for the above case, some estimates for functions of zero order, and a number of examples.
Reviewer:  J.Karlsson
Classif.:  * 41A20 Approximation by rational functions
30E10 Approximation in the complex domain

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