К вопросу о представл ении аналитических ф ункций рядами обобщенных эк спонент

Abstract

Let$f\left( z \right) = \sum\limits_0^\infty {\frac{{a_n }}{{n!}}} z^n$ be an entire function of exponential type and completely regular growth. Necessary and sufficient conditions forа _n are obtained in order that in each convex domainD each functionF(z) analytic inD be representable as a series

$$F\left( z \right) = \mathop \Sigma \limits_1^\infty A_n f\left( {\lambda _n z} \right).$$

((1))

Furthermore, conditions ona _n are obtained, under which each functionF(z) of order ϱ>1 and finite type be representable by such a series (1) that the sum of moduli of its terms possesses a definite indicator of growth.

This strengthens the author's results [7] and [6].