Boundary behavior and Cesàro means of universal Taylor series.

*(English)*Zbl 1103.30003Let \({\mathcal C}\) 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 \(j \in N\).

Reviewer: Peter Lappan (East Lansing)