Polynomial approximation on real-analytic varieties in

Authors:
John T. Anderson, Alexander J. Izzo and John Wermer

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1495-1500

MSC (2000):
Primary 32E30; Secondary 46J10

DOI:
https://doi.org/10.1090/S0002-9939-03-07263-0

Published electronically:
November 14, 2003

MathSciNet review:
2053357

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove: Let be a compact real-analytic variety in . Assume (i) is polynomially convex and (ii) every point of is a peak point for . Then . This generalizes a previous result of the authors on polynomial approximation on three-dimensional real-analytic submanifolds of .

**1.**H. Alexander and J. Wermer,*Several Complex Variables and Banach Algebras*, Third edition, Springer-Verlag, New York, 1998. MR**98g:32002****2.**J. T. Anderson and A. J. Izzo,*A Peak Point Theorem for Uniform Algebras Generated by Smooth Functions on a Two-Manifold*, Bull. London Math. Soc.**33**(2001), pp. 187-195. MR**2002j:32035****3.**J. T. Anderson, A. J. Izzo and J. Wermer,*Polynomial Approximation on Three-Dimensional Real-Analytic Submanifolds of*, Proc. Amer. Math. Soc.**129**(2001), pp. 2395-2402. MR**2002d:32021****4.**R. F. Basener,*On Rationally Convex Hulls*, Trans. Amer. Math. Soc.**182**(1973), pp. 353-381. MR**52:803****5.**A. Browder,*Introduction to Function Algebras*, Benjamin, New York, 1969. MR**39:7431****6.**H. Federer,*Geometric Measure Theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969. MR**41:1976****7.**M. Freeman,*Some Conditions for Uniform Approximation on a Manifold*, in: Function Algebras, F. Birtel (ed.), Scott-Foresman and Co., Chicago, 1966, pp. 42-60. MR**33:1758****8.**L. Hörmander and J. Wermer,*Uniform Approximation on Compact Subsets in*, Math. Scand.**23**(1968), pp. 5-21. MR**40:7484****9.**A. J. Izzo,*Failure of Polynomial Approximation on Polynomially Convex Subsets of the Sphere*, Bull. London Math. Soc.**28**(1996), pp. 393-397. MR**98d:32017****10.**R. Narasimhan,*Introduction to the Theory of Analytic Spaces*, Lecture Notes in Mathematics no. 25, Springer-Verlag, Berlin, 1966. MR**36:428****11.**A. G. O'Farrell, K. J. Preskenis, and D. Walsh,*Holomorphic Approximation in Lipschitz Norms*, in Proceedings of the Conference on Banach Algebras and Several Complex Variables, Contemporary Math., vol. 32, American Mathematical Society, Providence, RI, 1983. MR**86c:32015****12.**E. L. Stout,*The Theory of Uniform Algebras*, Bogden and Quigley, Tarrytown-on-Hudson, NY, 1971. MR**54:11066****13.**J. Wermer,*Polynomially Convex Disks*, Math. Ann.**158**(1965), pp. 6-10. MR**30:5158**

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Additional Information

**John T. Anderson**

Affiliation:
Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, Massachusetts 01610-2395

Email:
anderson@mathcs.holycross.edu

**Alexander J. Izzo**

Affiliation:
Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403

Address at time of publication:
Department of Mathematics, Brown University, Providence, RI 02912

Email:
aizzo@math.bgsu.edu, aizzo@math.brown.edu

**John Wermer**

Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island 02912

Email:
wermer@math.brown.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07263-0

Received by editor(s):
January 15, 2003

Published electronically:
November 14, 2003

Communicated by:
Mei-Chi Shaw

Article copyright:
© Copyright 2003
American Mathematical Society