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**Zentralblatt MATH**

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

**Zbl.No: ** 372.41008

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

**Title: ** Approximation by rational functions. (In English)

**Source: ** J. London Math. Soc., II. Ser. 15, 319-328 (1977).

**Review: ** This paper contains eight theorems on the rational approximation of e^{-x} . We cite one of them by way of an example: ''Let p(x) and q(x) be any polynomials of degress at most n-1 where n \geq 2. Then we have **|**|e^{-x}-\frac{p(x)}{q(x)}**|**|_{l_{oo}(N)} \geq \frac{(e-1)^{n}e^{-4n}2^{-7n}}{n(3+2\sqrt2)^{n-1}}.'', (N is the set of non-negative integers). Another theorems is a result of the same type for **|**|e^{-x}-\frac{p(x)}{q(x)}**|**|_{L_{oo}[0,1]}, with the restriction on p(x) that its coefficients are non-negative. It should have been mentioned that the rational function r_{m,n}(x) with denominator of degree m and numerator of degree n (not m), both defined by an integral, for which it is shown that, theorem 2,

**|**|e^{-x}-r_{m,n}(x)**|**|_{L_{oo}[0,1]} \leq \frac{m^{n}n^{n}}{(m,n)^{m+n}(m+n)!}, is in fact the Padé approximant of e^{-x}. From the various results applied during the proofs of the eight theorems we mention Lagrange's interpolation theorem, interpolation polynomials from the calculus of differences and a lemma of the second author which says that [p(x)]^{ 1/n } is concave on [a,b] when the polynomial p has degree at most n, has only real zeros and p(x) < 0 on [a,b].

**Reviewer: ** H.Jager

**Classif.: ** * 41A20 Approximation by rational functions

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