Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-08T12:20:51.363Z Has data issue: false hasContentIssue false
  • English
  • Français

Gérard-Sibuya's versus Majima's concept of asymptotic expansion in several variables

Part of: Approximations and expansions Asymptotic theory

Published online by Cambridge University Press:  09 April 2009

J. A. Hernández  and
J. Sanz
J. Sanz
Affiliation:
Depto. de Análisis Matemático Facultad de Ciencias Universidad de Valladolidc/ Prado de la Magdalena s/n 47005 ValladolidSpain e-mail: jaisla@am.uva.es e-mail: jsanzg@am.uva.es
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.

We give an example of a holomorphic function, admitting Gérard-Sibuya asymptotic expansion on a polysector of Cn, and such that none of its derivatives admits such an expansion. This motivates the study of the relationship between the concepts of asymptotic expansion in several variables respectively given by Gérard-Sibuya and Majima. For a function f, Majima's notion is proved to be equivalent, on the one hand, to the existence of Gérard-Sibuya asymptotic expansion for f and its derivatives, and on the other hand, to the boundedness of the derivatives of f on bounded proper subpolysectors of S.

MSC classification

Secondary: 34E05: Asymptotic expansions 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 1 , August 2001 , pp. 21 - 35
Copyright
Copyright © Australian Mathematical Society 2001

References

[GS]Gérard, R. and Sibuya, Y., ‘Étude de certains systèmes de Pfaff avec singularités’, in: Lecture Notes in Math. 172 (Springer, Berlin, 1979) pp. 131–288.Google Scholar
[Ha]Haraoka, Y., ‘Theorems of Sibuya-Malgrange type for Gevrey functions of several variables’, Funkcial. Ekvac. 32 (1989), 365–388.Google Scholar
[He]Hernández, J. A., Desarrollos asintóticos en polisectores. Problemas de existencia y unicidad (Ph. D. Thesis, University of Valladolid, 1994).Google Scholar
[HS]Hernández, J. A. and Sanz, J., ‘Constructive Borel-Ritt interpolation results for functions of several variables’, Asymptotic Anal. 24 (2000), 167–182.Google Scholar
[M1]Majima, H., ‘Analogues of Cartan's decomposition theorem in asymptotic analysis’, Funkcial. Ekvac. 26 (1983), 131–154.Google Scholar
[M2]Majima, H., Asymptotic analysis for integrable connections with irregular singular points, Lecture Notes in Math. 1075 (Springer, Berlin, 1984).CrossRefGoogle Scholar
[P]Poincaré, H., ‘Sur les intégrales des équations linéaires’, Acta Math. 8 (1886), 295–344.CrossRefGoogle Scholar
[R]Rudin, W., Análisis real y complejo (McGraw-Hill/Interamericana, Madrid, 1988),Google Scholar
translation into Spanish of Real and complex analysis, 3rd Edition (McGraw-Hill, New York, 1987).Google Scholar
[SG]Sanz, J. and Galindo, F., ‘On strongly asymptotically developable functions and the Borel-Ritt theorem’, Studia Math. 133 (1999), 231–248.Google Scholar
[W]Wasow, W., Asymptotic expansions for ordinary differential equations (John Wiley and Sons, New York, 1965).Google Scholar
[Z]Zurro, M. A., ‘A new Taylor type formula and ∞ extensions for asymptotically developable functions’, Studia Math. 123 (1997), 151–163.CrossRefGoogle Scholar