Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Polynomial approximation on real-analytic varieties in $\mathbf{C}^n$


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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove: Let $\Sigma$ be a compact real-analytic variety in $\mathbf{C}^n$. Assume (i) $\Sigma$ is polynomially convex and (ii) every point of $\Sigma$ is a peak point for $P(\Sigma)$. Then $P(\Sigma) = C(\Sigma)$. This generalizes a previous result of the authors on polynomial approximation on three-dimensional real-analytic submanifolds of $\mathbf{C}^n$.


References [Enhancements On Off] (What's this?)

  • 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 $\mathbf{C}^n$, 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 $\mathbf{C}^n$, 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32E30, 46J10

Retrieve articles in all journals with MSC (2000): 32E30, 46J10


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

American Mathematical Society