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Universal Taylor series, conformal mappings and boundary behaviour. (Séries de Taylor universelles, transformations conformes et comportement à la frontière.) (English. French summary) Zbl 1310.30050

Let \(\Omega\) be a simply connected domain \(\Omega\subset \mathbb{C}\), \(\Omega\neq \mathbb{C}\) and \(\zeta\in\Omega\). For \(f\in H(\Omega),\) we denote by \(S_N(f,\zeta)(z)\) the sequence of partial sums of the Taylor development of \(f\) with center \(\zeta\). It is well -known that generically for \(f\in H(\Omega)\) the following holds: for every compact set \(K\), \(K\cap\Omega=\emptyset\) with \(K^c\) connected, and every function \(h: K\to\mathbb{C}\) continuous on \(K\) and holomorphic in \(K^\circ\), there exists a strictly increasing sequence \(\lambda_n\in\{0,1,2,\dots\}\) such that \(\sup_{z\in K}|S_{\lambda_n}(f,\zeta)(z)-h(z)|\to 0\) as \(n\to+\infty\) [V. Nestoridis, in: Proceedings of the 3rd CMFT conference on computational methods and function theory 1997, Nicosia, Cyprus, 1997. Singapore: World Scientific. 421–430 (1999; Zbl 0942.30003)]. We denote by \(\mathcal{U}(\Omega,\zeta)\) the set of such universal functions. Since 1996 many authors have been investigating the properties of these universal Taylor series (see the references therein).
Let \(S=\{z\in\mathbb{C}:-1<\text{Re}z<1\}\) and \(\mathbb{D}=\{z\in\mathbb{C}:| z|<1\}.\) In the present paper, the author proves the two following main results :
- there is a function \(f\in\mathcal{U}(S,0)\) such that, for any conformal mapping \(F:\mathbb{D}\to S,\) \(f\circ F\notin \mathcal{U}(\mathbb{D},F^{-1}(0))\);
- if \(f\in \mathcal{U}(\mathbb{D},0),\) then for almost every point \(\xi,\) with \(| \xi|=1,\) \(f(\{z:| z-\xi|<\alpha (1-| z|)<\alpha t\})\) is dense in \(\mathbb{C}\) for all \(\alpha>1\) and \(0<t\leq 1\).
The first assertion shows that the universal approximation property is not conformally invariant. The second assertion shows that the classical universal Taylor series have extreme angular boundary behavior. To obtain these nice results, the author uses in a cunning way the potential theory. Finally the author shows that there exists \(f\) in \(\mathcal{U}(\mathbb{D},0)\) such that \(z\mapsto (z-1)f(z)\) does not belong to \(\mathcal{U}(\mathbb{D},0).\) This gives an answer to a question of Costakis.

MSC:

30K05 Universal Taylor series in one complex variable
30B30 Boundary behavior of power series in one complex variable; over-convergence
30E10 Approximation in the complex plane
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions

Citations:

Zbl 0942.30003
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References:

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