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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”.

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
Full Text: DOI
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