Interpolation and uniqueness results for entire functions

Abstract:

Let $K[\Omega ]$ denote the collection of entire functions of exponential type whose Borel transforms are analytic on ${\Omega ^c}$ (the complement of the simply connected domain taken relative to the sphere). Let $f$ be in $K[\Omega ]$ and set ${L_n}(f) = {(2\pi i)^{ - 1}}\smallint \Gamma {g_n}(\lambda )F(\lambda )d\lambda (n = 0,1, \ldots )$ where $F$ is the Borel transform of $f,\Gamma \subset \Omega$ is a simple closed contour chosen so that $F$ is analytic outside and on $\Gamma$ and each ${g_n}$ is in $H(\Omega )$ (the collection of functions analytic on $\Omega$). In what follows read ’the sequence of linear functionals $\{ {L_n}(f)\}$’ wherever the sequence of functions ’$\{ {g_n}\}$’ appears. Let $T$ denote a continuous linear operator from $H(\Omega )$ to $H(\Lambda )$ where $\Lambda$ is also a simply connected domain. The topologies on $H(\Omega )$ and $H(\Lambda )$ are those of uniform convergence on compact subsets of $\Omega$ (resp. $\Lambda$). The purpose of this paper is to consider uniqueness preserving operators, i.e., operators $T$ which have the property that $K[\Lambda ]$ is a uniqueness class for $\{ T({g_n})\}$ whenever $K[\Omega ]$ is a uniqueness class for $\{ {g_n}\}$, and to examine interpolation preserving operators, i.e., operators $T$ which have the property that $K[\Lambda ]$ interpolates the sequence of complex numbers $\{ {b_n}\}$ relative to $\{ T({g_n})\}$ whenever $K[\Omega ]$ interpolates $\{ {b_n}\}$ relatives to $\{ {g_n}\}$. Once some classes of uniqueness preserving operators and some classes of interpolation preserving operators have been found, we proceed to obtain new uniqueness and interpolation results from our knowledge of these operators and from previously known uniqueness and interpolation results. Operators which multiply by analytic functions and some differential operators are considered. Composition operators are studied and the results are used to extend the interpolation results for sequences of functions of the form $\{ {[W(\zeta )]^n}\}$ where $W$ is analytic and univalent on a simply connected domain.