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Qualitative rational approximation on plane compacta

Part of the Lecture Notes in Mathematics book series (LNM,volume 995)

Abstract

Let X be a compact subset of the complex plane. Let R(X) denote the space of all rational functions with poles off X. Let A(X) denote the space of all complex-valued functions on X that are analytic on the interior of X. Let A(X) be a Banach space of functions on X, with R(X)⊂A(X)⊂A(X). Consider the problems: (1) Describe the closure of R(X) in A(X). (2) For which X is R(X) dense in A(X)? There are many results on these problems, for various particular Banach spaces A(X). We offer a point of view from which these results may be viewed systematically.

Keywords

  • Rational Approximation
  • Open Disc
  • Banach Function Space
  • Fine Topology
  • Open Riemann Surface

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. T. Bagby, Quasi topologies and rational approximation, J. Functional Analysis, 10 (1972) 259–268.

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. J. Brennan, Approximation in the mean by polynomials on non-Caratheodory domains, Ark. Mat. (1977).

    Google Scholar 

  3. A. Browder, Lectures on Function Algebras, Benjamin, 1969.

    Google Scholar 

  4. A. M. Davie, Analytic capacity and approximation problems, Transactions AMS 171 (1972) 409–444.

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. T. W. Gamelin, Uniform Algebras, Prentice-Hall, 1969.

    Google Scholar 

  6. J. Garnett, Analytic Capacity and Measure, LNM 297, Springer, 1972.

    Google Scholar 

  7. P. M. Gauthier, Meromorphic uniform approximation on closed subsets of open Riemann surfaces, Approximation Theory and Functional Analysis, J. B. Prolla (ed.), North-Holland, 1979.

    Google Scholar 

  8. _____, Analytic approximation on closed subsets of open Riemann surfaces, to appear.

    Google Scholar 

  9. V. P. Havin, Approximation in the mean by analytic functions, Doklady Akad. Nauk SSSR, 178 (1968) 1025–1028.

    MathSciNet  Google Scholar 

  10. L. I. Hedberg, Approximation in the mean by analytic functions, Transactions AMS 163 (1972) 157–171.

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. _____, Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972) 299–319.

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. D. Marshall, Removable sets for bounded analytic functions, Contemporary Problems in Complex and Linear Analysis, S. N. Nikolskü (ed.), Nauka, 1980.

    Google Scholar 

  13. A. O’Farrell, Point derivations on an algebra of Lipschitz functions, Proceedings RIA 80A (1980) 23–39.

    MathSciNet  MATH  Google Scholar 

  14. _____, Localness of certain Banach modules, Indiana Univ. Math. J. 24 (1975) 1135–1141.

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. _____, Annihilators of rational modules, J. Functional Analysis 19 (1975) 373–389.

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. _____, Rational approximation in Lipschitz norms I, Proceedings RIA 77A (1977) 113–115.

    MathSciNet  MATH  Google Scholar 

  17. _____, Rational approximation in Lipschitz norms II, Proceedings RIA 79A (1979) 103–114.

    MathSciNet  MATH  Google Scholar 

  18. _____, Lip 1 rational approximation, Journal LMS (2) 11 (1975) 159–164.

    MathSciNet  MATH  Google Scholar 

  19. _____, Hausdorff content and rational approximation in fractional Lipschitz norms, Transactions AMS 228 (1977) 187–206.

    MathSciNet  CrossRef  MATH  Google Scholar 

  20. _____, Continuity properties of Hausdorff content, Journal LMS (2) 13 (1976) 403–410.

    MathSciNet  MATH  Google Scholar 

  21. _____, Estimates for capacities and approximation in Lipschitz norms, J.f.d. Reine u. Angew. Math. 311/312 (1979) 101–115.

    MathSciNet  MATH  Google Scholar 

  22. E. P. Smith, The Garabedian function of an arbitrary compact set, Pacific J. Math. 51 (1974), 289–300.

    MathSciNet  CrossRef  MATH  Google Scholar 

  23. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton 1970.

    Google Scholar 

  24. A. G. Vitushkin, Analytic capacity of sets and problems of approximation theory, Russian Math. Surveys 22 (1967) 139–200.

    MathSciNet  CrossRef  MATH  Google Scholar 

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© 1983 Springer-Verlag

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O’Farrell, A.G. (1983). Qualitative rational approximation on plane compacta. In: Blei, R.C., Sidney, S.J. (eds) Banach Spaces, Harmonic Analysis, and Probability Theory. Lecture Notes in Mathematics, vol 995. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0061890

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  • DOI: https://doi.org/10.1007/BFb0061890

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12314-9

  • Online ISBN: 978-3-540-40036-3

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