Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T06:05:44.440Z Has data issue: false hasContentIssue false
  • English
  • Français

The Usual Behaviour of Rational Approximations

Published online by Cambridge University Press:  20 November 2018

Peter B. Borwein
Peter B. Borwein*
Affiliation:
Dalhousie University Halifax, Nova Scotia Canada B3H 4H8
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Questions concerning the convergence of Padé and best rational approximations are considered from a categorical point of view in the complete metric space of entire functions. The set of functions for which a subsequence of the mth row of the Padé table converges uniformly on compact subsets of the complex plane is shown to be residual.

The speed of convergence of best uniform rational approximations and Padé approximations on the unit disc is compared. It is shown that, in a categorical sense, it is expected that subsequences of these approximants will converge at the same rate.

Likewise, it is expected that the poles of certain sequences of best uniform rational approximations wil be dense in the entire plane.

Keywords

41 A20A21A25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 3 , 01 September 1983 , pp. 317 - 323
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Baker, G. A. Jr, and Graves-Morris, P. R., Convergence of the Fade Table, J. Math. Anal. App. 57 (1977), 323-339.Google Scholar
2. Borwein, P., On Padé and Best Rational Approximations, Canad. Math. Bull. 26 (1) (1983) 50-57.Google Scholar
3. Chui, C. K., Recent results on Fade Approximants and related problems, in Lorentz, G. G., Schumaker, L. L. (eds.), Proceedings, Conference on Approximation Theory, Academic Press, (1976).Google Scholar
4. Gragg, W. B., The Padé table and its applications to certain algorithms of numerical analysis, SIAM Review, 14 (1972), 1-62.Google Scholar
5. Gončar, A A., Properties of functions related to their rate of approximability by rational functions, Amer. Math. Soc. Transi. (2) 91 (1970), 99-128.Google Scholar
6. Saff, E. B., and Varga, R. S., On the zeros and poles of Padé approximants to ez , Numer. Math., 25 (1975), 1-14.Google Scholar
7. Saff, E. B., On the degree of best rational approximation to the exponential function, J. Approx. Theory, 9 (1973), 97-101.Google Scholar