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)ne-4n2-7n}{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 rm,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-rm,n(x)||L_{oo[0,1]} \leq \frac{mnnn}{(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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