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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 386.30020

**Autor: ** Erdös, Paul; Newman, Donald J.; Reddy, A.R.

**Title: ** Rational approximation. II. (In English)

**Source: ** Adv. Math. 29, 135-156 (1978).

**Review: ** Let \pi_{m} denote the class of all real polynomials of degree at most m and \pi_{m,n} denote the collection of all rational functions r_{m,n}(x) = \frac{p(x)}{q(x)}, p in \pi_{m}, q in \pi_{n}. Let \lambda_{m,n}\equiv\lambda_{m,n}(f^{-1}) = **inf**_{r_{m,n} in \pi_{m,n}}**|**|\frac1{f(x)}-\gamma_{m,n}(x)**|**|_{l_{oo}[0,oo]} where f, given by f(z) = **sum**_{k = 0}^{oo}a_{k}z^{k}, is an entire function with all non-negative coefficients. In Part I [*P.Erdös* and *A.R.Reddy*, Adv. Math. 21, 78-109 (1976; Zbl 334.00019)] , the authors mainly reviewed and proved certain results concerning \lambda_{0,n}. In the present paper, which contains 22 theorems, the authors devote themselves to show that for certain classes of entire functions the error obtained by rational functions of degree n in approximating on [0,oo) under the uniform norm is much smaller than the error obtained by recipocals of polynomials of degree n. For example, they show in Theorem 10 that if f is an entire function of order \rho(1 \leq \rho < oo), type \tau and lower type \omega(0 < \omega \leq \tau < oo), then for every polynomial P_{n}(x) of degree n and all large n, there exist positive constants a and b for which **|**|\frac{x+1}{f(x)}-\frac 1{P_{n}(x)}**|**|_{L_{oo}[0,oo)} \geq a\exp((-bn^{1-1/3\rho}) whereas in Theorem 17 they establish that for such functions there is some \beta (0 < \beta < 1) such that \lambda_{1,n}**(**\frac{1+x}{f(x)}**)** \leq \beta^{n}. It has also been shown that for certain entire functions, for example f(z) = e^{e^{z}}, there is little difference between the errors abtained by rational functions and the errors obtained by recipocals of polynomials. Incidentally, the following interesting results has also been obtained: **lim**_{n ––> oo}**[**\lambda_{0,n}(xe^{-x})**]**^{ ½ log n} = e^{-1} = **lim**_{n ––> oo}**[**\lambda_{0,n}((1+x)e^{-x})**]**^{{1/(2n)}^{2/3}}.

**Reviewer: ** O.P.Juneja

**Classif.: ** * 30E10 Approximation in the complex domain

41A20 Approximation by rational functions

41A25 Degree of approximation, etc.

**Citations: ** Zbl 334.30019

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag