Abstract
Let G be a simply connected, bounded domain on the plane with the boundary Γ and let P(Γ) be a uniform closure of polynomials on Γ. It is shown that the Rudin-Carleson Theorem about analytic extensions from zero measure boundary sets is valid for P(Γ) if and only if G is a Carathéodory domain and \({\bar G}\) does not separate the plane. These conditions are also equivalent to maximality of P(Γ) in C(Γ).
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References
E. Bishop, The structure of certain measures, Duke Math. J. 25 no.2 (1958), 283–289.
A. Browder, Introduction of Function Algebras, W. A. Benjamin, New York, 1969.
E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets, Cambridge Univ. Press, Cambridge, 1966.
A. M. Davie, Dirichlet algebras of analytic functions, J. Functional Analysis 6 (1970), 348–356.
A. A. Dovgoshei, The F. and M. Riesz’ theorem and Carathéodory domains, Analysis Mathematica, 21 (1995), 165–175.
S. Eilenberg, An invariance theorem for subsets of Sn, Bull. Amer. Math. Soc. 47 (1941), 73–75.
S. D. Fisher, Function Theory on Planar Domains, A Second Course in Complex Analysis, John Wiley & Sons. New York etc., 1983.
D. Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser Verlag, Basel, Boston, Stuttgart, 1980.
T. W. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969.
W. Hayman and P. Kennedy, Subharmonic Functions, vol. I, Academic Press. London etc., 1976.
J. L. Kelley, General Topology, D. Van Nostrand Company, Princeton, 1965.
K. Kuratawski, Topology, vol. II, Academic Press, New York, 1968.
G. M. Leibowitz, Lectures on Complex Function Algebras, Scott, Foresman and Company, Glenview, Illinois, 1970.
W. Rudin, Functional Analysis, Second Edition, McGraw-Hill, New York etc., 1991.
W. Rudin, Analyticity and the maximum modulus principle, Duke Math. J. 20 (1953), 449–457.
J. L. Walsh, The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions, Bull. Amer. Soc. 35 (1929), 499–544.
J. Wermer, Banach algebras and analytic functions, in: H. Busemann (ed.), Advances in Mathematics 1, Academic press, New York and London, 1961.
J. Wermer, On algebras of continuous functions, Proc. Amer. Math. Soc. 4 (1953), 866–869.
K. Yoneyama, Theory of continuous sets of points, Tohoku Math. Jour. 11 (1917), 43–158.
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Dovgoshey, O. Certain Characterizations of Carathéodory Domains. Comput. Methods Funct. Theory 5, 489–503 (2006). https://doi.org/10.1007/BF03321112
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DOI: https://doi.org/10.1007/BF03321112
Keywords
- Caratheodory domains
- harmonic measure
- polynomial approximation
- peak sets
- interpolation sets
- maximal subalgebras