On uniform tangential approximations by lacunary power series on curves in the complex plane (Q1896869)

Let \(D_R\) denote the \(\text{disc} \{|z |< R\}\) for \(0 < R \leq \infty\). For an appropriate piecewise smooth Jordan \(\text{arc} \Gamma_0\) in \(D_R\) joining the origin and a point of \(\partial D_R\), put \(E_m = \cup^{m - 1}_{k = 0} \Gamma_k\), where \(\Gamma_k = \{z \in D_R : z = w \exp (2 \pi ik/m)\), \(w \in \Gamma_0\}\). The authors prove: For an appropriate lacunary sequence \(Q\) and for \(\varepsilon > 0\) and \(f\) arbitrary continuous functions on \(E_m\), there exists a function \(g\) holomorphic in \(D_R\), \(g(z) = \sum_{n \in Q} g_n z^n\), such that \(|f(z) - g(z) |< \varepsilon (z)\), \(\forall z \in E_m\).