A Levinson-Sjöberg type theorem. Applications

A generalization of the well-known Levinson-Sjöberg theorem is obtained for a family of analytic functions $f$ that have estimates of the form $|f(z)|\le M(\operatorname{dist}(z,\gamma))$ outside an arc $\gamma$, where $M$ is a decreasing function on $(0,\infty)$ that is unbounded in a neighbourhood of the origin. Applications to questions of quasianalyticity for Carleman classes are indicated as well as to the completeness of a system of exponentials on arcs, to analytic continuation and to representation by Dirichlet series.
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