Universal overconvergence and Ostrowski-gaps.

*(English)*Zbl 1099.30001Let \(O\) 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.

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.

Reviewer: Peter Lappan (East Lansing)