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

Stout, Edgar

doi:10.1090/S0002-9939-06-08250-5

Holomorphic approximation on compact, holomorphically convex, real-analytic varieties
HTML articles powered by AMS MathViewer

by Edgar Lee Stout PDF

Proc. Amer. Math. Soc. 134 (2006), 2303-2308 Request permission

Abstract:

Every continuous function on a compact, holomorphically convex, real-analytic subset of $\mathbb {C}^N$ can be approximated uniformly by functions holomorphic on the set.

References

Patrick Ahern and Walter Rudin, Totally real embeddings of $S^3$ in $\textbf {C}^3$, Proc. Amer. Math. Soc. 94 (1985), no. 3, 460–462. MR 787894, DOI 10.1090/S0002-9939-1985-0787894-7
John T. Anderson, Alexander J. Izzo, and John Wermer, Polynomial approximation on real-analytic varieties in $\mathbf C^n$, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1495–1500. MR 2053357, DOI 10.1090/S0002-9939-03-07263-0
Frank T. Birtel, Some holomorphic function algebras, Papers from the Summer Gathering on Function Algebras (Aarhus, 1969) Aarhus Univ., Matematisk Inst., Aarhus, 1969, pp. 11–18. MR 0254598
F.T. Birtel, Algebras of Holomorphic Functions: 40 Lectures, Tulane University Mathematics Department, New Orleans, 1972.
Jan-Erik Björk, Holomorphic convexity and analytic structures in Banach algebras, Ark. Mat. 9 (1971), 39–54. MR 385170, DOI 10.1007/BF02383636
Klas Diederich and John E. Fornaess, Pseudoconvex domains with real-analytic boundary, Ann. of Math. (2) 107 (1978), no. 2, 371–384. MR 477153, DOI 10.2307/1971120
Julien Duval and Nessim Sibony, Polynomial convexity, rational convexity, and currents, Duke Math. J. 79 (1995), no. 2, 487–513. MR 1344768, DOI 10.1215/S0012-7094-95-07912-5
Reese Harvey and R. O. Wells Jr., Compact holomorphically convex subsets of a Stein manifold, Trans. Amer. Math. Soc. 136 (1969), 509–516. MR 235158, DOI 10.1090/S0002-9947-1969-0235158-8
F. Reese Harvey and R. O. Wells Jr., Holomorphic approximation and hyperfunction theory on a $C^{1}$ totally real submanifold of a complex manifold, Math. Ann. 197 (1972), 287–318. MR 310278, DOI 10.1007/BF01428202
C. Denson Hill and Geraldine Taiani, Families of analytic discs in $\textbf {C}^{n}$ with boundaries on a prescribed CR submanifold, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 327–380. MR 501906
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, DOI 10.1112/blms/28.4.393
Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR 0217337
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., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 187–194. MR 769507, DOI 10.1090/conm/032/769507
P. J. de Paepe, Homomorphism spaces of algebras of holomorphic functions, Pacific J. Math. 66 (1976), no. 1, 211–220. MR 442282
A. Selvaggi Primicerio and G. Taiani, Families of analytic discs with boundary on a nongeneric CR submanifold, Ann. Mat. Pura Appl. (4) 126 (1980), 233–251 (1981) (Italian, with English summary). MR 612361, DOI 10.1007/BF01762509