Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T08:07:39.902Z Has data issue: false hasContentIssue false
  • English
  • Français

BOUNDED POLYNOMIALS AND HOLOMORPHIC MAPPINGS BETWEEN CONVEX SUBRINGS OF *$\mathbb{C}$

Published online by Cambridge University Press:  08 February 2018

ADEL KHALFALLAH  and
SIEGMUND KOSAREW
ADEL KHALFALLAH
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS DHAHRAN31261, SAUDI ARABIAE-mail:khelifa@kfupm.edu.sa
SIEGMUND KOSAREW
Affiliation:
INSTITUT FOURIER, UNIVERSITÉ GRENOBLE ALPES 100 RUE DES MATHS 38610 GIÈRES, FRANCEE-mail:siegmund.kosarew@ujf-grenoble.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Using convex subrings of *$\mathbb{C}$, a nonstandard extension of $\mathbb{C}$, we define several kinds of complex bounded polynomials and we provide their associated analytic functions obtained by taking the quasistandard part.

Keywords

46F3026E3503H0546S1012J25nonstandard analysisinternal polynomialsgeneralized holomorphicity
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 372 - 384
Copyright
Copyright © The Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bourbaki, N., Commutative Algebra, Addison-Wesley, Paris, Hermann, 1972.Google Scholar
Colombeau, J. F., New Generalized Functions and Multiplication of Distributions, North-Holland, Amsterdam, 1984.Google Scholar
Colombeau, J. F., Elementary Introduction to New Generalized Functions, North-Holland, Amsterdam, 1985.Google Scholar
Goldblatt, R., Lectures on the Hyperreals, Springer-Verlag, New York, 1998.Google Scholar
Hurd, A. E. and Loeb, P. A., An Introduction to Nonstandard Real Analysis, Academic Press, Orlando, FL, 1985.Google Scholar
Khalfallah, A. and Kosarew, S., Complex spaces and nonstandard schemes. Journal of Logic and Analysis, vol. 2 (2010), paper 9, pp. 1–60.Google Scholar
Khelif, A. and Scarpalezos, D., Zeros of generalized holomorphic functions. Monatshefte für Mathematik, vol. 149 (2006), pp. 323–335.Google Scholar
Lindstrom, T., An invitation to nonstandard analysis, Nonstandard Analysis and its Applications (Cutland, N., editor), Cambridge University Press, New York, 1988, pp. 1–105.Google Scholar
Lightstone, A. H. and Robinson, A., Nonarchimedean Fields and Asymptotic Expansions, North-Holland, Amsterdam, 1975.Google Scholar
Oberguggenberger, M. and Todorov, T., An embedding of Schwartz distributions in the algebra of asymptotic functions. International Journal of Mathematics and Mathematical Sciences, vol. 21 (1998), no. 3, pp. 417–428.Google Scholar
Oberguggenberger, M., Pilipovic, S., and Valmorin, V., Global representatives of Colombeau holomorphic generalized functions. Monatshefte für Mathematik, vol. 151 (2007), pp. 67–74.Google Scholar
Robinson, A., Nonstandard Analysis, North Holland, Amsterdam, 1966.Google Scholar
Stroyan, K. D. and Luxemburg, W. A. J., Introduction to the Theory of Infinitesimals, Academic Press, New York, 1976.Google Scholar
Todorov, T. D., Lecture Notes: Non-Standard Approach to J.F. Colombeau’s Theory of Generalized Functions, arXiv:1010.3482.Google Scholar