A necessary and sufficient condition for uniform approximation by certain rational modules

Carmona Doménech, J. J.

doi:10.1090/S0002-9939-1982-0671221-7

A necessary and sufficient condition for uniform approximation by certain rational modules
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by J. J. Carmona Doménech PDF

Proc. Amer. Math. Soc. 86 (1982), 487-490 Request permission

Abstract:

Let $X$ be a compact subset of ${\mathbf {C}}$ with empty interior and let $g$ be a complex function of class ${C^2}$ in a neighborhood of $X$. For $Z = \left \{ {z \in X|\partial g(z)/\partial \bar z = 0} \right \}$, we prove that $R(X) + gR(X)$ is uniformly dense in $C(X)$ if and only if $R(Z) = C(Z)$.

References

Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125
Robert B. Burckel, An introduction to classical complex analysis. Vol. 1, Pure and Applied Mathematics, vol. 82, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 555733, DOI 10.1007/978-3-0348-9374-9
Anthony G. O’Farrell, Annihilators of rational modules, J. Functional Analysis 19 (1975), no. 4, 373–389. MR 0380428, DOI 10.1016/0022-1236(75)90063-4
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
Tavan Trent and James Li Ming Wang, Uniform approximation by rational modules on nowhere dense sets, Proc. Amer. Math. Soc. 81 (1981), no. 1, 62–64. MR 589136, DOI 10.1090/S0002-9939-1981-0589136-0
James Li Ming Wang, Approximation by rational modules on nowhere dense sets, Pacific J. Math. 80 (1979), no. 1, 293–295. MR 534719, DOI 10.2140/pjm.1979.80.293
Lawrence Zalcmann, Analytic capacity and rational approximation, Lecture Notes in Mathematics, No. 50, Springer-Verlag, Berlin-New York, 1968. MR 0227434, DOI 10.1007/BFb0070657