Universality and Cesàro summability
A function \(f\) that is holomorphic on an open set \(\Omega\) is said to be universal at the point \(\zeta\in\Omega\) if for every compact set \(K\subset\mathbb{C}\setminus \Omega\) with connected complement and for every function \(g\) continuous on \(K\) and holomorphic in the interior of \(K\) there exists a strictly increasing sequence \((\lambda_n)\) of natural numbers such that \[ \displaystyle\lim_{n\rightarrow +\infty} \sup_{z\in K}| S_{\lambda_n}(f,\zeta)(z)-g(z)|=0, \] where \(S_N(f,\zeta)\) denotes the \(N\)th-partial sum of the Taylor development of \(f\) at the point \(\zeta.\) It is well known that there exist universal functions when \(\Omega\neq \mathbb{C}\) is a simply connected domain in \(\mathbb{C}\). Moreover the Taylor series with center \(\zeta\in \Omega\) of such universal functions are not \((C,a)\)-summable for every \(a\geq 1\) at any point of the boundary \(\partial \Omega\) of \(\Omega\) [\textit{A. Melas} and \textit{V. Nestoridis}, Adv. Math. 157, No. 2, 138--176 (2001; Zbl 0985.30023)]. On the other hand, \textit{W. Gehlen} et al. [Complex Variables, Theory Appl. 41, No. 1, 81--90 (2000; Zbl 1020.30004)] proved that universal Taylor series do not exist in bounded non-simply connected domains. First, in the opposite direction, the authors prove the following result announced in [C. R. Math. Acad. Sci. Paris 347, No. 7--8, 363--367 (2009; Zbl 1218.30003)]: Let \(\Omega\subset\mathbb{C}\) be a domain, \(\Omega\neq\mathbb{C}\), \(\zeta\in\Omega\), \[ R=\text{dist}(\zeta,\Omega^c)\in (0,+\infty),\; \; C(\zeta,R)=\{z\in\mathbb C\:\;|\zeta-z|=R\},\;\;\text{and}\;\; J(\Omega,\zeta)=\Omega^c\cap C(\zeta,R). \] Then there exists \(f\in H(\Omega)\) such that the sequence \(S_N(f,\zeta)(z)\), \(N=0,1,\dots\), approximates any polynomial uniformly on each compact set \(K\subset J(\Omega,\zeta)\) with \(\mathbb{C}\setminus K\) connected. The set of such universal functions is a \(G_\delta\)-dense subset of \(H(\Omega)\) and contains a dense vector subspace minus the zero element. Then, they give sufficient conditions on the compact set \(K\) defined in the previous statement and \(z_0\in K\) which guarantee that the above universal Taylor series is not \((C,a)\)-summable for every \(a>-1\) at \(z_0.\) To do this, they use Rogosinski's formula and its technical improvements. Conversely if \(\Omega=\{z\in\mathbb{C};\;| z|<1\}\) and \(K\) is a finite subset of \(\{z\in\mathbb{C};\;| z|\geq 1\},\) they give an example of a Taylor series \(\sum_{n\geq 0}c_nz^n\) in \(\Omega\) which is universal with respect to \(K\) and \((C,1)\)-summable at every point \(z\in K\) with \(| z|=1\). The paper is very well written and provides lots of examples.
scientific article; zbMATH DE number 6124795
scientific article