Hostname: page-component-6bf8c574d5-pdxrj Total loading time: 0 Render date: 2025-03-07T03:24:38.348Z Has data issue: false hasContentIssue false
  • English
  • Français

Weierstrass Division in Quasianalytic Local Rings

Published online by Cambridge University Press:  20 November 2018

C. L. Childress
C. L. Childress*
Affiliation:
Case Western Reserve University, Cleveland, Ohio
Rights & Permissions [Opens in a new window]

Extract

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.

In this paper we consider the problem of extending the Weierstrass division theorem to quasianalytic local rings of germs of functions of k real variables which properly contain the local ring of germs of analytic functions. After some background material (§ 2) and some technical preliminaries (§ 3), we show by examples that the so-called generic division theorem fails in such rings if k ≧ 1 and that the Weierstrass division theorem fails in such rings if k ≧ 2 (§ 4).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 5 , 01 October 1976 , pp. 938 - 953
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Chollet, A. M., Zéros dans les classes de gevrey de type analytique, Bull. Sc. Math., 2e Série, 96 (1972), 65–82.Google Scholar
2. Bochner, S. and Martin, W. T., Several complex variables (Princeton University Press, Princeton, 1948).Google Scholar
3. Davis, P. J., Interpolation and approximation (Blaisdell Publishing Co., New York, 1963).Google Scholar
4. Ehrenpreis, L., Fourier analysis in several complex variables (John Wiley and Sons, Inc., New York, 1970).Google Scholar
5. Grothendieck, A., Espaces vectoriels topologiques, Sociedad de Mathemâtica de Sâo Paulo, Sào Paulo, 1964.Google Scholar
6. Hôrmander, L., An introduction to complex analysis in several variables (North Holland, 1973). 7. Katznelson, Y., An introduction to harmonic analysis (John Wiley and Sons, Inc., New York, 1968).Google Scholar
8. Rudin, W., Real and complex analysis (McGraw-Hill Co., Inc., New York, 1966).Google Scholar
9. Wall, C. T. C., Introduction to the preparation theorem, Proceedings of Liverp∞l Singularities- Symposium I, ed. Wall, C. T. C. (Springer-Verlag, Berlin, 1971).Google Scholar