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Carleman Approximation by Entire Functions on the Union of Two Totally Real Subspaces of Cn

Published online by Cambridge University Press:  20 November 2018

Per E. Manne*
Affiliation:
Department of Mathematics, Norwegian School of Economics and Business Administration N-5035 Bergen-Sandviken Norway
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Abstract

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Let L 1, L 2 ⊂ C n be two totally real subspaces of real dimension n, and such that L 1L 2 = {0}. We show that continuous functions on L 1L 2 allow Carleman approximation by entire functions if and only if L 1L 2 is polynomially convex. If the latter condition is satisfied, then a function f:L 1L 2 —> C such that f|L i Ck(Li), i = 1,2, allows Carleman approximation of order k by entire functions if and only if f satisfies the Cauchy-Riemann equations up to order k at the origin.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Frih, E. M., Approximation holomorphe globale sur Vunion de deux sous-espaces totalement réels maximaux de Cn, Canad. Math. Bull. 34(1991), 215219.Google Scholar
2. Manne, P., Carle man Approximation on Totally Real Submanifolds of a Complex Manifold, In: Several Complex Variables: Proceedings of the Mittag-Leffler Institute, 1987-1988, (ed. J. E. Fornaess), Princeton University Press, New Jersey, 1993.Google Scholar
3. Range, R. M. and Siu, Y.-T., Ck Approximation by Holomorphic Functions and -Closed Forms on Ck Submanifolds of a Complex Manifold, Math. Ann. 210(1974), 105122.CrossRefGoogle Scholar
4. Stout, E. L. and Zame, W. R., Totally Real Imbeddings and the Universal Covering Spaces of Domains of Holomorphy: Some Examples, Manuscripta Math. 50(1985), 2948.Google Scholar
5. Weinstock, B., On the Polynomial Convexity of the Union of Two Maximal Totally Real Subspaces of Cn, Math. Ann. 282(1988), 131138.Google Scholar
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