On divergence of series of exponents representing functions regular in convex polygons (Q1905116)

Let \(M\) be a convex polygon with vertices at points \(\gamma_1, \dots, \gamma_N\) such that \(0 \in M\) and let \(E^1 (M)\) denote the Smirnov space on \(M\). For \(f \in E^1 (M)\) let \(\Sigma (f)\) be the exponential series of \(f\). The author presents the following results: (1) There exists a function \(f \in E^1(M)\) such that \(|S_n(f) |_{E^1 (M)} \to \infty\), where \(S_n(f)\), \(n \in \mathbb{N}\), denote the partial sums of \(\Sigma (f)\). (2) There exists a function \(f \in E^1 (M)\) such that \(\Sigma (f)\) is divergent almost everywhere on \(\partial M\). (3) There exists a function \(f \in {\mathcal O} (M) \cap {\mathcal C} (\overline M)\) such that \(\Sigma (f)\) is convergent on \(\partial M\) but not uniformly. (4) There exists a function \(f \in {\mathcal O} (M) \cap {\mathcal C} (\overline M)\) such that \(\Sigma (f)\) is divergent at all non-corner points from \([\gamma_j, \gamma_{j + 1}]\), \(j = 1, \dots, N - 1\).