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 \pim denote the class of all real polynomials of degree at most m and \pim,n denote the collection of all rational functions rm,n(x) = \frac{p(x)}{q(x)}, p in \pim, q in \pin. Let \lambdam,n\equiv\lambdam,n(f-1) = infr_{m,n in \pim,n}||\frac1{f(x)}-\gammam,n(x)||l_{oo[0,oo]} where f, given by f(z) = sumk = 0ooakzk, 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 \lambda0,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 Pn(x) of degree n and all large n, there exist positive constants a and b for which ||\frac{x+1}{f(x)}-\frac 1{Pn(x)}||L_{oo[0,oo)} \geq a\exp((-bn1-1/3\rho) whereas in Theorem 17 they establish that for such functions there is some \beta (0 < \beta < 1) such that \lambda1,n(\frac{1+x}{f(x)}) \leq \betan. It has also been shown that for certain entire functions, for example f(z) = ee^{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:

limn ––> oo[\lambda0,n(xe-x)] ½ log n = e-1 = limn ––> oo[\lambda0,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