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Boundary behavior and Cesàro means of universal Taylor series. (English) Zbl 1103.30003
Let $${\mathcal C}$$ denote the complex plane, and let $$f$$ be a function holomorphic in the unit disc $$D$$. Let $$N$$ denote the set of all natural numbers. For $$\zeta \in D$$, we can write $$f(z)$$ as a Taylor series with center at $$\zeta$$ in the form $$f(z) = \sum a_{n} (z - \zeta)^{n}$$, where the series converges uniformly in each compact disc centered at $$\zeta$$ and contained in $$D$$. Let $$S_{n}(f, \zeta)(z)$$ denote the $$n$$-th partial sum of the power series for $$f$$ with center at $$\zeta$$. We say that $$f$$ belongs to the class $$U(D,\zeta)$$ if for each compact set $$K \subset {\mathcal C}$$, where both $$K \cap D = \emptyset$$ and $$K$$ has connected complement, and for each function $$h$$ continuous on $$K$$ and holomorphic in the interior of $$K$$, there exists a sequence $$\{\lambda_{n}\}$$ in $$N$$ such that the sequence of functions $$\{S_{\lambda_{n}}(f, \zeta)(z)\}$$ converges uniformly to $$h(z)$$ on $$K$$. Further, a function holomorphic in $$D$$ belongs to the class $$U(D)$$ if for each compact subset $$K \subset {\mathcal C}$$, where both $$K \cap D = \emptyset$$ and $$K$$ has connected complement, and for each function $$h$$ continuous on $$K$$ and holomorphic in the interior of $$K$$, there exists a sequence $$\{\lambda_{n}\}$$ in $$N$$ such that for each compact subset $$A \subset D$$, $\sup_{\zeta \in A} \sup_{z \in K} | S_{\lambda_{n}}(f, \zeta)(z) - h(z)| \to 0 \text{ as } n \to \infty \;.$ The author improves a result of D. Armitage and G. Costakis [Constructive Approximation 24, No.1, 1–15 (2006; Zbl 1098.30003)] by showing that if $$f \in U(D)$$ there exists a residual subset $$G$$ of the unit circle $$T$$ such that the set $$\{f^{(n)}(rz): 0 < r < 1\}$$ is unbounded for each $$z \in G$$ and each $$n \geq 0$$. (Armitage and Costakis prove this for each $$n \geq 1$$.) Let $$\sigma^{j}_{n}(f, \zeta)(z)$$ be the $$n$$-th Cesàro mean of order $$j$$ of the Taylor series for $$f$$ with center at $$\zeta$$. Let $$U_{\text{Ces} (j)}(D,\zeta)$$ denote the class of functions holomorphic in $$D$$ such that the definition of $$f \in U(D, \zeta)$$ is satisfied by using $$\sigma^{j}_{\lambda_{n}}(f, \zeta))(z)$$ in place of $$S_{\lambda_{n}}(f, \zeta)(z)$$. Also, let $$U_{\text{Ces} (j)}(D)$$ denote the class of functions $$f$$ holomorphic in $$D$$ such that the definition of $$f \in U(D)$$ is satisfied by using $$\sigma^{j}_{\lambda_{n})(f, \zeta})(z)$$ in place of $$S_{\lambda_{n}}(f, \zeta)(z)$$. Finally, let $U_{\text{Ces}}(D, \zeta) = \bigcap_{j=0}^{\infty} U_{\text{Ces}(j)}(D, \zeta) \quad \text{and} \quad U_{\text{Ces}}(D) = \bigcap_{j=0}^{\infty} U_{\text{Ces}(j)}(D).$ The author proves that $U(D) = U_{\text{Ces}}(D) = U_{\text{Ces}}(D, \zeta) = U_{\text{Ces}(j)}(D, \zeta) = U_{\text{Ces}(j)}(D)$ for each $$\zeta \in D$$ and $$j \in N$$.

##### MSC:
 30B40 Analytic continuation of functions of one complex variable 30D40 Cluster sets, prime ends, boundary behavior 40G05 Cesàro, Euler, Nörlund and Hausdorff methods
##### Keywords:
universal series; Ostrowski gaps; overconvergence
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