Geometric condition for universal interpolation in $\hat {\mathcal {E}}’$

Abstract:

It is known that if $h$ is an entire function of exponential type and $Z(h) = {\{ {z_k}\} _{k = 1}}$ with $|h’({z_k})| \geqslant \varepsilon \exp (- c|{z_k}|)$ for constants $\epsilon$, $C$ independent of $k$, then $\{ {z_k}\} _{k = 1}^\infty$ is a universal interpolation sequence. That is, given any sequence of complex numbers $\{ {a_k}\} _{k = 1}^\infty$ such that $|{a_k}| \leqslant A \exp (B|{z_k}|)$ for constants $A,B$ independent of $K$ then there exists $g$ of exponential type such that $g({z_k}) = {a_k}$. This note is concerned with finding geometric conditions which make $\{ {z_k}\} _{k = 1}^\infty$ a universal interpolation sequence for various spaces of entire functions. For the space of entire functions of exponential type a necessary and sufficient condition for $\{ {z_k}\} _{k = 1}^\infty$ to be a universal interpolation sequence is that $\int _0^{|{z_k}|} {n({z_k},t) dt/t \leqslant C|{z_k}| + D,k = 1} , 2,\ldots$, where $n({z_k},t)$ is the number of points of $\{ {z_k}\} _{k = 1}^\infty$ in the disc of radius $t$ about ${z_k}$, excluding ${z_k}$, and $C,D$ are constants independent of $k$. Results for the space $\hat {\mathcal {E}}^\prime = \{ f\;{\text {entire}}||f(z)| \leqslant A\;\exp [B|\operatorname {Im} z| + B\log (1 + |z|^{2})]\}$ are given but the theory is not as complete as for the above example.