Polynomial approximation on real-analytic varieties in 𝐂ⁿ

Anderson, John; Izzo, Alexander; Wermer, John

doi:10.1090/S0002-9939-03-07263-0

Polynomial approximation on real-analytic varieties in $\mathbf {C}^n$
HTML articles powered by AMS MathViewer

by John T. Anderson, Alexander J. Izzo and John Wermer PDF

Proc. Amer. Math. Soc. 132 (2004), 1495-1500 Request permission

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

Herbert Alexander and John Wermer, Several complex variables and Banach algebras, 3rd ed., Graduate Texts in Mathematics, vol. 35, Springer-Verlag, New York, 1998. MR 1482798
John T. Anderson and Alexander J. Izzo, A peak point theorem for uniform algebras generated by smooth functions on two-manifolds, Bull. London Math. Soc. 33 (2001), no. 2, 187–195. MR 1815422, DOI 10.1112/blms/33.2.187
John T. Anderson, Alexander J. Izzo, and John Wermer, Polynomial approximation on three-dimensional real-analytic submanifolds of $\textbf {C}^n$, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2395–2402. MR 1823924, DOI 10.1090/S0002-9939-01-05911-1
Richard F. Basener, On rationally convex hulls, Trans. Amer. Math. Soc. 182 (1973), 353–381. MR 379899, DOI 10.1090/S0002-9947-1973-0379899-1
Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Michael Freeman, Some conditions for uniform approximation on a manifold, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 42–60. MR 0193538
L. Hörmander and J. Wermer, Uniform approximation on compact sets in $C^{n}$, Math. Scand. 23 (1968), 5–21 (1969). MR 254275, DOI 10.7146/math.scand.a-10893
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
Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0423083
J. Wermer, Polynomially convex disks, Math. Ann. 158 (1965), 6–10. MR 174968, DOI 10.1007/BF01370392