Universal transforms of the geometric series under generalized Riesz methods (Q1884431)

The authors consider generalized Riesz methods \((R,p,M)\) of summability. They obtain two results on approximation of holomorphic functions. For example. Suppose we have: (i) an open set \(O\subset\mathbb C\) with simply connected components and \(\mathbb{D}\subset O\), \(1\notin O\); (ii) a set \(F\subset O^{c}\) which has the property \(E\); (iii) a sequence \(\{P_n\}\subset\mathbb{C}\setminus\{0\}\) with \(P_n\to\infty\). Let \(\Phi\) be a function in \(H(O)\) with \(\Phi| _{\mathbb D}=1/(1-z)\). Then there exist sequences \(\{p_\nu\}\subset\mathbb C\) and \(\{m_n\}\subset\{0,1,2,\dots\}\) such that we have \(P_n=\sum_{\nu=0}^{m_n}p_\nu\) \((n\in\mathbb N)\) and such that \[ \frac1{P_n}\sum\limits_{\nu=0}^{m_n}p_\nu\sum\limits_{\mu=0}^{\nu}z^\mu \Longrightarrow\Phi(z)\,. \] Furthermore, the authors employ these assertions to establish a strong result about approximation of Lebesgue-measurable functions.