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A characterization of the growth of analytic functions by means of polynomial approximation. (English) Zbl 0554.32015

Let K be a compact balanced subset of \({\mathbb{C}}^ N\) such that the Siciak extremal function \(\Phi_ K\) is continuous, where \(\Phi_ K(z)=\sup \{| p(z)|^{1/n}:\) p polynomial of degree \(\leq n,\) \(\sup_{K}| p| \leq 1,\quad n\geq 1\}.\) For \(1<R<\infty,\) let \(K_ R=\{z\in {\mathbb{C}}^ N:\quad \Phi_ K(z)<R\}\) and \({\mathcal O}(R)=set\) of holomorphic functions in \(K_ R\) and not continuable to any \(K_{R'}\), \(R'>R.\) For \(g\in {\mathcal O}(R)\) define its order of growth \(\rho\) by \[ \rho = \limsup_{r\to R}[\log^+\log^+\sup \{| g(z)|: \Phi_ K(z)=r\}/(-\log (1-r/R))]. \] The author proves: Theorem. Let f be a function defined and bounded in K. For each \(n\geq 1\), let \(t_ n\) denote its n-th Chebychev polynomial of best approximation in K. Then f is the restriction to K of a function \(g\in {\mathcal O}(R)\) of order \(\rho\), \(0<\rho <\infty,\) if and only if \[ \limsup_{n\to \infty}(\log^+\log^+(R^ n\sup_{K}| f-t_ n|)/\log n)=\rho /(\rho +1). \] There are similar results for the restriction of a function of a given order \(\rho\) and type \(\sigma\). The notion of Chebychev polynomial appears in a paper by J. Siciak [see Trans. Am. Math. Soc. 105, 322-357 (1962; Zbl 0111.081)].
Reviewer: C.A.Berenstein

MSC:

32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
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