Extension of entire functions of completely regular growth and right inverse to the~operator of representation of analytic functions by quasipolynomial series

S. N. Melikhov

Sbornik. Mathematics, Tome 191 (2000) no. 7, pp. 1049-1073

Résumé

Let $L$ be an entire function of one complex variable that has exponential type, completely regular growth, and whose conjugate diagram is equal to the sum of the closure of a bounded convex domain $G$ and a convex compact subset $K$ of $\mathbb C$. Criteria ensuring that the operator $R$ of the representation of analytic functions in $G$ by quasipolynomial series with zeros of the function $L$ as exponents has a continuous linear right inverse are established. These criteria are stated in terms of conformal maps of the unit disc onto the domain $G$ and of the exterior of the closed unit disc onto the exterior of $K$, and of extensions of the original function $L$ to an entire function $Q$ of two complex variables whose absolute value satisfies certain (upper) estimates. An analogue of the Leont'ev interpolation function defined by this extension $Q$ is used to obtain formulae for the continuous linear right inverse to the representation operator $R$.