Theorem concerning analytic continuation

Abstract

A. I. Markushevich obtained the following representation of a function in its holomorphicity star with a sequence {m _v }, for which m _v+1/m _v→∞:

Here it is proved that this condition is necessary; more precisely,$\overline {\mathop {\lim }\limits_{v \to \infty } } \frac{{m_{v + 1} }}{{m_v }} = \infty$. This result is derived from certain properties of over-convergent power series.