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Some applications of the Robin function to multivariable approximation theory. (English) Zbl 0896.31003
Author’s abstract: “Let $$E\subset \mathbb{C}^n$$ be compact, regular, and polynomially convex with pluricomplex Green function $$V_E$$. Given a sequence of polynomials $$\{p_j\}_{j=1,2,\dots}$$, the first result is a condition for $$(\varlimsup_{j\to\infty} (\log | p_j(z)|/ \deg(p_j)))^*$$ to equal $$V_E$$ on $$\mathbb{C}^n- E$$. The condition involves the Robin function of $$E$$ and the highest order homogeneous terms of the $$p_j$$ and generalizes one-variable results of Blatt-Saff. A second result gives a necessary and sufficient condition for $$f$$, the uniform limit of polynomials on $$E$$, to extend holomorphically to $$E_R= \{z\mid V_E(z)< \log R\}$$ for $$R>1$$. The condition involves highest order homogeneous terms of best approximating polynomials to $$f$$ and the Robin function of $$E$$ and extends results of Szczepański”.

##### MSC:
 31C10 Pluriharmonic and plurisubharmonic functions 41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX) 32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables 41A10 Approximation by polynomials
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##### References:
 [1] Berenstein, C.; Gay, R., Complex variables. an introduction, Springer graduate texts in mathematics, 125, (1991), Springer-Verlag New York/Berlin [2] Blatt, H.-P.; Saff, E.B., Behaviour of zeros of polynomials of near best approximation, J. approx. theory, 46, 323-344, (1986) · Zbl 0605.41026 [3] Bedford, E.; Taylor, B.A., Plurisubharmonic functions with logarithmic singularities, Ann. inst. Fourier (Grenoble), 38, 133-171, (1988) · Zbl 0626.32022 [4] Fornaess, J.E.; Stensones, B., Lectures on counterexamples in several complex variables, (1987), Princeton Univ. Press Princeton · Zbl 0626.32001 [5] Klimek, M., Pluripotential theory, (1991), Oxford Univ. Press Oxford · Zbl 0742.31001 [6] N. Levenberg, Capacities in Several Complex Variables, University of Michigan, 1984 [7] Nivoche, S., The pluricomplex Green function, capacitative notions, and approximation problems in $$C$$^n, Indiana univ. math. J., 44, 489-510, (1995) · Zbl 0846.31008 [8] Pleśniak, W., Ln, Bull. Polish acad. sci., 32, 647-651, (1984) [9] Sadullaev, A., Plurisubharmonic measures and capacities on complex manifolds, Russian math. surveys, 36, 61-119, (1981) · Zbl 0494.31005 [10] Shabat, B.V., Introduction to complex analysis, part II, functions of several variables, Translations of mathematical monographs, 10, (1991), American Math. Society Providence [11] Shafarevich, I.R., Basic algebraic geometry, (1977), Springer-Verlag New York/Berlin · Zbl 0362.14001 [12] Siciak, J., Extremal plurisubharmonic functions in $$C$$^n, Ann. Polish math., 39, 175-211, (1981) · Zbl 0477.32018 [13] J. Siciak, A remark on Tchebysheff polynomials in $$C$$^n, Jagiellonian University, 1996 · Zbl 0951.32011 [14] Szczepański, J., Indicators of growth of polynomials of best approximation to holomorphic functions on compacta in $$C$$^n, J. approx. theory, 76, 233-245, (1994) · Zbl 0796.41022 [15] Taylor, B.A., An estimate for an extremal plurisubharmonic function on $$C$$^n, Lecture notes in math., 1028, (1983), Springer-Verlag New York/Berlin, p. 318-328 [16] Nguyen, T.V.; Zeriahi, A., Famille de polynômes presque partout bornées, Bull. sci. math. (2), 107, 81-91, (1983) · Zbl 0523.32011 [17] Wójcik, A.P., On zeroes of polynomials of best approximation to holomorphic andC∞, Mh. math., 105, 75-81, (1988) · Zbl 0639.41005 [18] Zeriahi, A., Capacité, constante de tchebysheff et polynômes orthogonaux associés a un compact de $$C$$^n, Bull. sci. math. (2), 109, 325-335, (1985) · Zbl 0583.31006
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