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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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