Holomorphic approximation on compact, holomorphically convex, real-analytic varieties

Author:
Edgar Lee Stout

Journal:
Proc. Amer. Math. Soc. **134** (2006), 2303-2308

MSC (2000):
Primary 32E30

DOI:
https://doi.org/10.1090/S0002-9939-06-08250-5

Published electronically:
February 2, 2006

MathSciNet review:
2213703

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Abstract | References | Similar Articles | Additional Information

Abstract: Every continuous function on a compact, holomorphically convex, real-analytic subset of can be approximated uniformly by functions holomorphic on the set.

**1.**P. Ahern and W. Rudin,*Totally real embeddings of in*, Proc. Amer. Math. Soc., 94, 1985, pp. 460-462. MR**0787894 (86g:32031)****2.**John T. Anderson, Alexander J. Izzo, and John Wermer,*Polynomial approximation on real-analytic varieties in 𝐂ⁿ*, Proc. Amer. Math. Soc.**132**(2004), no. 5, 1495–1500. MR**2053357**, https://doi.org/10.1090/S0002-9939-03-07263-0**3.**Frank T. Birtel,*Some holomorphic function algebras*, Papers from the Summer Gathering on Function Algebras (Aarhus, 1969) Matematisk Inst., Aarhus Univ., Aarhus, 1969, pp. 11–18. MR**0254598****4.**F.T. Birtel,*Algebras of Holomorphic Functions: 40 Lectures*, Tulane University Mathematics Department, New Orleans, 1972.**5.**Jan-Erik Björk,*Holomorphic convexity and analytic structures in Banach algebras*, Ark. Mat.**9**(1971), 39–54. MR**0385170**, https://doi.org/10.1007/BF02383636**6.**Klas Diederich and John E. Fornaess,*Pseudoconvex domains with real-analytic boundary*, Ann. Math. (2)**107**(1978), no. 2, 371–384. MR**0477153****7.**Julien Duval and Nessim Sibony,*Polynomial convexity, rational convexity, and currents*, Duke Math. J.**79**(1995), no. 2, 487–513. MR**1344768**, https://doi.org/10.1215/S0012-7094-95-07912-5**8.**Reese Harvey and R. O. Wells Jr.,*Compact holomorphically convex subsets of a Stein manifold*, Trans. Amer. Math. Soc.**136**(1969), 509–516. MR**0235158**, https://doi.org/10.1090/S0002-9947-1969-0235158-8**9.**F. Reese Harvey and R. O. Wells Jr.,*Holomorphic approximation and hyperfunction theory on a 𝐶¹ totally real submanifold of a complex manifold*, Math. Ann.**197**(1972), 287–318. MR**0310278**, https://doi.org/10.1007/BF01428202**10.**C.D. Hill and G. Taiani,*Families of analytic discs in with boundaries on a prescribed CR submanifold*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), (5) 1978, pp. 327-380. MR**0501906 (80c:32023)****11.**Alexander J. Izzo,*Failure of polynomial approximation on polynomially convex subsets of the sphere*, Bull. London Math. Soc.**28**(1996), no. 4, 393–397. MR**1384828**, https://doi.org/10.1112/blms/28.4.393**12.**Raghavan Narasimhan,*Introduction to the theory of analytic spaces*, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR**0217337****13.**A.G. O'Farrell, K.J. Preskenis, and D. Walsh,*Holomorphic approximation in Lipschitz norms*, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983), Contemp. Math., (32), 187-194. Amer. Math. Soc., Providence, RI, 1984. MR**0769507 (86c:32015)****14.**P. J. de Paepe,*Homomorphism spaces of algebras of holomorphic functions*, Pacific J. Math.**66**(1976), no. 1, 211–220. MR**0442282****15.**A. Selvaggi Primicerio, and G. Taiani,*Famiglie di dischi analitici con bordo su sottovarietà CR non generiche*, Ann. Mat. Pura Appl. (4), (126), 1980, pp. 233-251. MR**0612361 (82j:32047)**

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Additional Information

**Edgar Lee Stout**

Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195

Email:
stout@math.washington.edu

DOI:
https://doi.org/10.1090/S0002-9939-06-08250-5

Keywords:
Holomorphic approximation,
holomorphically convex sets.

Received by editor(s):
June 8, 2004

Received by editor(s) in revised form:
March 2, 2005

Published electronically:
February 2, 2006

Communicated by:
Mei-Chi Shaw

Article copyright:
© Copyright 2006
American Mathematical Society