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 $R$.

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