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A theorem of A. G. Vitushkin

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Abstract

An important condition for the coincidence of the algebras R(E) and C(E) on a compactum

was found by A. G. Vitushkin. In this note we give a simple proof of Vitushkin's theorem.

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Literature cited

  1. A. G. Vitushkin, “Analytic capacity of a set in problems of approximation theory,” Usp. Mat. Nauk,22, No. 6, 141–199 (1967).

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  2. A. G. Vitushkin, “Conditions on a set which are necessary and sufficient for the possibility of uniform approximation by analytic (or rational) functions of any function which is continuous on this set,” Dokl. Akad. Nauk SSSR,128, No. 1, 17–21 (1959).

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  3. T. Gamelin, Uniform Algebras [Russian translation], Moscow (1973).

  4. L. Schwartz, Theorie des Distributions, Vol. 1, Paris (1950).

  5. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (1971).

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Authors
  1. N. A. Shirokov
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 178–181, 1984.

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Shirokov, N.A. A theorem of A. G. Vitushkin. J Math Sci 31, 2746–2748 (1985). https://doi.org/10.1007/BF02107261

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  • DOI: https://doi.org/10.1007/BF02107261

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