Geodesic
Approximation by polynomials in the mean with respect to area
Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 411-421 
S. O. Sinanyan. Approximation by polynomials in the mean with respect to area. Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 411-421. http://geodesic.mathdoc.fr/item/SM_1970_11_3_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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[2] A. I. Markushevich, Teoriya analiticheskikh funktsii, Gostekhizdat, Moskva–Leningrad, 1950

[3] O. J. Farrell, “On approximation to on anlytic function by polynomials”, Bull. Amer. Math. Soc., 40 (1934), 908–914 | DOI | Zbl

[4] C. H. Mergelyan, “O polnote sistem analiticheskikh funktsii”, Uspekhi matem. nauk, VIII:4(56) (1953), 3–63 | MR

[5] S. O. Sinanyan, “Approksimatsiya analiticheskimi i polinomami v srednem po ploschadi”, Matem. sb., 69(111) (1966), 546–578 | Zbl

[6] S. O. Sinanyan, “Svoistvo edinstvennosti analiticheskikh funktsii na mnozhestvakh bez vnutrennikh tochek”, Sib. matem. zh., 6:6 (1965), 1365–1381 | Zbl

[7] V. P. Khavin, “Approksimatsiya analiticheskimi funktsiyami v srednem”, DAN SSSR, 178:5 (1968), 1025–1028 | Zbl

[8] J. Wermer, “Banach algebras and analytic functions”, Advan. Math., 1:1 (1961) | MR | Zbl

[9] A. G. Vitushkin, “Analiticheskaya emkost mnozhestv v zadachakh teorii priblizhenii”, Uspekhi matem. nauk, XXII:6(138) (1967), 142–199