Abstract:
This paper extends the concept of polynomial hull to arbitrary bounded sets of a complex plane, when an integral metric (relative to area) is considered. It is proved that for the completeness of the set of polynomials in a given closed or open set according to an integral metric, it is a necessary condition that the corresponding “polynomial hull” coincide with the interior of the set under consideration. The sufficiency of this condition is proved for various classes of sets. Using the notion of analytic p-capacity of sets, we obtain a full description of the compacta for which the polynomials are complete.
Bibliography: 9 titles.
\Bibitem{Sin70}
\by S.~O.~Sinanyan
\paper Approximation by polynomials in the mean with respect to area
\jour Math. USSR-Sb.
\yr 1970
\vol 11
\issue 3
\pages 411--421
\mathnet{http://mi.mathnet.ru/eng/sm3460}
\crossref{https://doi.org/10.1070/SM1970v011n03ABEH001299}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=274776}
\zmath{https://zbmath.org/?q=an:0224.30053}
Linking options:
https://www.mathnet.ru/eng/sm3460
https://doi.org/10.1070/SM1970v011n03ABEH001299
https://www.mathnet.ru/eng/sm/v124/i3/p444
This publication is cited in the following 5 articles:
James E. Brennan, Progress in Mathematics, 4, Complex Approximation, 1980, 32
James E. Brennan, “Point evaluations, invariant subspaces and approximation in the mean by polynomials”, Journal of Functional Analysis, 34:3 (1979), 407
James E. Brennan, “Approximation in the mean by polynomials on non-Carathéodory domains”, Ark Mat, 15:1-2 (1977), 117
James E. Brennan, “Invariant subspaces and weighted polynomial approximation”, Ark Mat, 11:1-2 (1973), 167
V. G. Maz'ya, V. P. Havin, “Use of $(p,l)$-capacity in problems of the theory of exceptional sets”, Math. USSR-Sb., 19:4 (1973), 547–580
Citation in format AMSBIB
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