Uniform rational approximation
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- by Li Ming Yang PDF
- Proc. Amer. Math. Soc. 123 (1995), 201-206 Request permission
Abstract:
Let K be a compact subset of the complex plane $\mathbb {C}$, and let $P(K)$ and $R(K)$ be the closures in $C(K)$ of polynomials and rational functions with poles off K, respectively. Suppose that $R(K) \ne C(K),\lambda$ is a nonpeak point for $R(K)$, and g is continuous on $\mathbb {C}$ and ${C^1}$ in a neighborhood of $\lambda$. Then $P(K)g + R(K)$ is not dense in $C(K)$. In fact, our proof shows that there are a lot of smooth functions which are not in the closure of $P(K)g + R(K)$.References
- Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
- James E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991), no. 3, 477–507. MR 1109351, DOI 10.2307/2944317
- James E. Thomson, Uniform approximation by rational functions, Indiana Univ. Math. J. 42 (1993), no. 1, 167–177. MR 1218711, DOI 10.1512/iumj.1993.42.42009
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 201-206
- MSC: Primary 30H05; Secondary 30E10, 41A20, 46J10
- DOI: https://doi.org/10.1090/S0002-9939-1995-1242112-7
- MathSciNet review: 1242112