On a theorem of P. L. Uljanov

Abstract:

It is shown that if $c\left ( n \right ) = o\left ( 1 \right )$, $\left | n \right | \to \infty$, and ${\sum _{\left | n \right |}}_{ < \infty }\left | {{\Delta ^m}c\left ( n \right )} \right | < \infty$, for some integer $m \geqslant 1$, then the series ${\sum _{\left | n \right |}}_{ < \infty }c\left ( n \right ){e^{\operatorname {int} }}$ converges to some $f \in {L^p}\left ( {\mathbf {T}} \right )$ for any $0 < p < 1 / m$.