Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T06:34:11.220Z Has data issue: false hasContentIssue false
  • English
  • Français

Perfect approximation of functions

Published online by Cambridge University Press:  17 April 2009

A.J. van der Poorten
A.J. van der Poorten
Affiliation:
The University of New South Wales, Kensington, New South Wales.
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.

There are only isolated instances of vectors of functions for which it is possible to obtain an explicit expression for the remainder functions obtained upon approximating by polynomials in the manner described by Mahler in his paper “Perfect systems”, Compositio Math. 19 (1968). We display appropriate identities and point to a pattern amongst these which suggests we should not expect convenient generalization to wider classes of functions. Proofs of perfectness do not require laborious computation but are immediate from the identities given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 5 , Issue 1 , August 1971 , pp. 117 - 126
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Baker, A., “A note on the Padé table”, K. Nederl. Akad. Wetensch. Proc. Ser. A 69 (1966), 596–601.CrossRefGoogle Scholar
[2]Coates, John, “On the algebraic approximation of functions. I, II, III”, K. Nederl. Akad. Wetensch. Proc. Ser. A 69 (1966), 421–434, 435–448, 449–461.CrossRefGoogle Scholar
[3]Coates, John, “On the algebraic approximation of functions. IV”, K. Nederl. Akad. Wetensch. Proc. Ser. A 70 (1967), 205–212.CrossRefGoogle Scholar
[4]Gelfond, A.O. and Linnik, Yu.V., Elementary methods in analytic number theory (Translated by Amiel Feinstein. Revised and edited by Mordell, L.J.. Rand McNally, Chicago, 1965; George Allen and Unwin, London, 1966).Google Scholar
[5]Mahler, K., “Perfect systems”, Compositio Math. 19 (1968), 95–166.Google Scholar
[6]Mahler, Kurt, “Zur Approximation der Exponentialfunktion und des Logarithmus, I, II”, J. reine angew. Math. 166 (1931), 118–136, 137–150.Google Scholar
[7]Mahler, Kurt, “Ein Beweis des Thue-Siegelschen Satzes über die Approximation algebraischer Zahlen für binomische Gleichungen”, Math. Ann. 105 (1931), 267–276.CrossRefGoogle Scholar
[S]Mahler, K., “On the approximation of logarithms of algebraic numbers” Philos. Trans. Roy. Soc. London Ser. A 245 (1953), 371–398.Google Scholar
[9]Pólya, G. und Szegö, G., Aufgaben und Lehrsätze aus der Analysis, Zweiter Band, 2te Aufl. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 20, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1951).Google Scholar
[10]Poorten, A.J. van der, “Generalizations of Turán's main theorems on lower bounds for sums of powers”, Bull. Austral. Math. Soc. 2 (1970), 15–37.CrossRefGoogle Scholar
[11]van der Poorten, A.J., “A generalization of Turán's main theorems to binomials and logarithms”, Bull. Austral. Math. Soc. 2 (1970), 183–183.CrossRefGoogle Scholar
[12]van der Poorten, A.J., “Generalised Vandermonde determinants”, (to appear).Google Scholar