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On the localization of singularities of lacunary power series. (English) Zbl 1123.30001

In this paper, the existence of singularities of lacunary power series on prescribed open boundary arcs \(\gamma\) of the circle of convergence is studied. Let \(f\) be a normalized analytic element, that is, \(f(z) \equiv \sum_{n=0}^\infty f_n z^n\) is a power series with the centre at the origin and radius of convergence \(1\) (so \(\limsup_{n \to \infty} | f_n| ^{1/n} = 1\)), representing a holomorphic function on the unit disc \(D\) having at least one singular point on \(\partial D_1\). Set \(P_+ := \mathbb N_0 \setminus P_0\), where \(\mathbb N_0 = \{0,1,2,\dots \}\) and \(P_0\) is the set \(\{n \in\mathbb N_0:f_n = 0\}\) of gaps of the series defining \(f\). The influence of the gaps on the length of the above arcs \(\gamma\) is expressed in terms of newly introduced integral densities.
Specifically, let \(P \subset \mathbb N_0\) and \(Q\) be a subsequence of \(\mathbb N := \{1,2,\dots \}\). The integral minimal density of \(P\) with respect to \(Q\) is defined as
\[ D_* (P,Q) := \lim_{s \to 0} \left( \liminf_{n \in Q} {1 \over s} \int_0^s {c((P \setminus \{n\}) \cap [(1-t)n,(1+t)n]) \over 2tn} \, dt \right), \] where \(c(A)\) denotes the cardinality of \(A\). The authors prove that if \(\gamma \subset \partial D_1\) is an open arc and \(f\) satisfies \(2 \pi D_* (P,Q) <\) length\((\gamma )\) for some subsequence \(Q\) with \(\lim_{n \in Q} | f_n| ^{1/n} = 1\), then \(f\) has a singular point on \(\gamma\). This implies that \(f\) has a singular point on each closed arc \(\overline{\gamma} \subset \partial D_1\) with length\((\gamma ) = 2 \pi D_* (P,Q)\). Since \(D_* (P,Q) \leq \Delta^* (P) =\) [the maximal density of \(P\)] \(:= \lim_{s \to 1} \limsup_{r \to \infty} {c(P \cap [0,r]) - c(P \cap [0,\mu r]) \over (1-\mu )r}\), one obtains as a consequence the Fabry-Pólya theorem on gaps.
For the proof of their main result, the authors establish the following necessary and sufficient condition (in terms of the so called “coefficient functions”) for the analytic continuation across boundary arcs: If \(\alpha \in [0,\pi )\), then the open arc \(\{e^{i \theta}: \alpha < \theta < 2\pi - \alpha\}\) is an arc of regularity of the series \(f\) if and only if there is an entire function \(\varphi\) of exponential type satisfying \(\varphi (n) = f_n\) \((n \in\mathbb N_0)\), \(h_\varphi (0) = 0\) and \(\limsup_{\theta \to 0} h_\varphi (\theta )/| \theta | \leq \alpha\), where \(h_\varphi (\theta )\) is the indicator function \(\limsup_{r \to \infty} (r^{-1} \log | \varphi (r e^{i \theta})| )\) of \(\varphi\).

MSC:

30B10 Power series (including lacunary series) in one complex variable
30B30 Boundary behavior of power series in one complex variable; over-convergence
30B40 Analytic continuation of functions of one complex variable
30D20 Entire functions of one complex variable (general theory)
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References:

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