Invariance of the $L$-regularity of compact sets in $\textbf {C}^{N}$ under holomorphic mappings
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- by W. Pleśniak
- Trans. Amer. Math. Soc. 246 (1978), 373-383
- DOI: https://doi.org/10.1090/S0002-9947-1978-0515544-4
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Abstract:
The property for a polynomially convex compact set E in ${C^N}$ that the Siciak extremal function ${\Phi _E}$ be continuous or, equivalently, that E satisfy some Bernstein type inequality, is proved to be invariant under a large class of holomorphic mappings with values in ${C^M}(M \leqslant N)$ including all open holomorphic mappings. Local specifications of this result are also given.References
- M. S. Baouendi and C. Goulaouic, Approximation of analytic functions on compact sets and Bernstein’s inequality, Trans. Amer. Math. Soc. 189 (1974), 251–261. MR 352789, DOI 10.1090/S0002-9947-1974-0352789-7
- Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
- Bengt Josefson, On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on $\textbf {C}^{n}$, Ark. Mat. 16 (1978), no. 1, 109–115. MR 590078, DOI 10.1007/BF02385986 N. S. Landkof, Foundations of modern potential theory, Moscow, 1966 (Russian).
- W. Pleśniak, On superposition of quasianalytic functions, Ann. Polon. Math. 26 (1972), 73–84. MR 308443, DOI 10.4064/ap-26-1-73-84
- W. Pleśniak, Quasianalytic functions in the sense of Bernstein, Dissertationes Math. (Rozprawy Mat.) 147 (1977), 66. MR 427674 —, Remarques sur une généralisation de l’inégalité de S. Bernstein, C. R. Acad. Sci. Paris Sér. A 284 (1977), 1211-1213.
- Józef Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322–357. MR 143946, DOI 10.1090/S0002-9947-1962-0143946-5 —, Extremal plurisubharmonic functions in ${C^N}$, Proceedings of the First Finnish-Polish Summer School in Complex Analysis at Podlesice (Łódź 1977), University of Łóodź, pp. 115-152.
- Joseph Siciak and Nguyen Thanh Van, Remarques sur l’approximation polynomiale, C. R. Acad. Sci. Paris Sér. A 279 (1974), 95–98 (French). MR 348145
- V. P. Zaharjuta, Extremal plurisubharmonic functions, orthogonal polynomials, and the Bernšteĭn-Walsh theorem for functions of several complex variables, Ann. Polon. Math. 33 (1976/77), no. 1-2, 137–148 (Russian). MR 444988
Bibliographic Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 246 (1978), 373-383
- MSC: Primary 32E30; Secondary 32E20
- DOI: https://doi.org/10.1090/S0002-9947-1978-0515544-4
- MathSciNet review: 515544