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Universal overconvergence and Ostrowski-gaps. (English) Zbl 1099.30001
Let $$O$$ be a domain in the complex plane $${\mathcal C}$$, and let $$f$$ be a function holomorphic in $$O$$. For a point $$\zeta \in O$$ and a positive integer $$n$$, let $S_{n}(f, \zeta)(z) = \sum_{j=0}^{n} \frac {f^{(j)}(\zeta)} {j!} (z - \zeta)^{j}$ be the $$n$$–th partial sum of the Taylor expansion of $$f$$ around $$\zeta$$. Let $$H(O)$$ denote the set of all functions holomorphic on $$O$$ using the topology of uniform convergence on compact sets. For a point $$\zeta_{0} \in O$$ and a set $B \subset \{z \in {\mathcal C}: | z - \zeta_{0}| \geq \text{dist}(\zeta_{0}, \partial O)\},$
we say that the sequence $$\{S_{n}(f, \zeta_{0})\}$$ is universally overconvergent in $$B$$ if for each compact subset $$K \subset B$$ with $$K^{c}$$, the complement of $$K$$, connected and for each $$g:K \to {\mathcal C}$$ that is continuous on $$K$$ and holomorphic in $$K^{o}$$, the interior of $$K$$, there exists a sequence $$n_{k}$$ of natural numbers such that $\sup_{z \in K} | S_{n_{k}}(f, \zeta_{0})(z) - g(z)| \to 0.$
Let $$U(O, \zeta_{0}, B)$$ denote the class of all functions $$f \in H(O)$$ such that $$\{S_{n}(f, \zeta_{0}\}$$ is universally overconvergent in $$B$$. Further, let $$U(O, B)$$ denote the collection of all functions $$f \in H(O)$$ such that for each compact subset $$K \subset B$$ with $$K^{c}$$ connected and for each $$g:K \to {\mathcal C}$$ that is continuous on $$K$$ and holomorphic in $$K^{o}$$, there exists a sequence $$n_{k}$$ of natural numbers such that $\sup_{\zeta \in L} \sup_{z \in K}{ | S_{n_{k}}(f, \zeta)(z)} - g(z)| \to 0$ for each compact subset $$L \subset O$$. It is known that if $$O$$ is simply connected then $$U(O, O^{c})$$ is a dense $$G_{\delta}$$ subset of $$H(O)$$, and if $$O$$ is not simply connected then $$U(O, O^{c}) = \emptyset$$. For a power series $$\sum a_{j}(z - \zeta_{0})^{j}$$ with radius of convergence $$r > 0$$, we say that the series has Ostrowski-gaps if there exist sequences $$\{p_{j}\}$$ and $$\{q_{j}\}$$ of positive integers with $1 \leq p_{1} < q_{1} \leq p_{2}< q_{2} \leq \cdots \leq p_{j} < q_{j} \leq \cdots$
such that both
$p_{j}/q_{j} \to 0 \;\;\text{ and } \lim_{m \to \infty, m \in I} | a_{m}| ^{1/m}, \;\text{ where } I = \bigcup_{j=1}^{\infty} [p_{j} + 1, q_{j}].$
For a domain $$O$$, a point $$\zeta \in O$$, and a function $$f \in H(O)$$, and a set
$B \subset \{z \in {\mathcal C}: | z - \zeta_{0}| \geq \text{dist}(\zeta_{0}, \partial B) \},$
we say that the sequence $$\{S_{n}(f, \zeta_{0})$$ is Ostrowski-gap universal in $$B$$ if for each compact subset $$K \subset B$$ with $$K^{c}$$ connected and for each function $$g:K \to {\mathcal C}$$ continuous on $$K$$ and holomorphic on $$K^{o}$$, the sequence $$n_{j}$$ which gives universal overconvergence in $$B$$ can be chosen so that $$\{S_{n_{j}}\}$$ has Ostrowski-gaps. Let $$U_{Ost}(O, \zeta_{0}, B)$$ denote the class of all functions $$f \in H(O)$$ such that $$\{S_{n}(f, \zeta_{0})\}$$ is Ostrowski-gap universal. Finally, a set $$S$$ is said to be non-thin at a point $$w \in {\mathcal C}$$ if $$w$$ is an accumulation point of $$S$$ and if
$\limsup_{z \to w, z \in S \setminus \{w\}} u(z) = u(w)$
for each function $$u$$ subharmonic in a neighborhood of $$w$$, and $$S$$ is called non-thin at $$\infty$$ if the set $$\{1/z: z \in S, z \neq 0\}$$ is non-thin at the point $$0$$. The authors prove that if $$O$$ is a domain and $$\zeta_{0} \in O$$, then
(i) if $$O$$ is simply connected then $$U(O, \zeta_{0}. O^{c}) = U_{Ost}(O, \zeta_{o},O^{c}) = U(O, O^{c})$$ and every $$f \in U(O, \zeta_{0}, O^{c})$$ is not holomorphically extendable beyond $$O$$.
(ii) If $$O$$ is not simply connected and if $$O^{c}$$ is non-thin at $$\infty$$, then $$U(O, \zeta_{0}, O^{c}) = \emptyset$$.
(iii) If $$\overline{O}^{c}$$, the complement of the closure of $$O$$, is non-thin at $$\infty$$, then $$U(O, \zeta_{0}, \overline{O}^{c}) = U_{Ost}(O, \zeta_{0},\overline{O}^{c}) = U(O, \overline{O}^{c})$$. Further, if $$f \in U(O, \zeta_{o}, \overline{O}^{c})$$ then the natural domain of $$f$$ is a subset of $$\overline{O}^{o}$$.
(iv) If $$\overline{O}^{c}$$ is non-thin at $$\infty$$ and if $$O^{c}$$ has a bounded component with nonempty interior, then $$U(O, \zeta_{0}, \overline{O}^{c}) = \emptyset$$.
Some similar results are proved for meromorphic functions using a series representation similar to a Laurent series.

##### MSC:
 30B10 Power series (including lacunary series) in one complex variable 30B30 Boundary behavior of power series in one complex variable; over-convergence
##### Keywords:
universally overconvergent; Ostrowski-gap universal
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