On classes of everywhere divergent power series

Abstract

We prove the everywhere divergence of series $$ \sum_{n=0}^\infty a_n e^{i\rho_n}e^{inx}, \quad\text{and}\quad \sum_{n=0}^\infty {(-1)}^{[\rho_n]}a_n \cos nx, $$ for sequences an and ρn satisfying some extremal conditions. These results generalize some well known examples of everywhere divergent power and trigonometric series.