On analytic sampling theory

Let (H,〈·,·〉H) be a complex, separable Hilbert space with orthonormal basis {xn}n=1∞ and let Ω be a domain in C, the field of complex numbers. Suppose K is a H-valued function defined on Ω. For each x∈H, define fx(z)=〈K(z),x〉H and let H denote the collection of all such functions fx. In this paper, we endow H with a structure of a reproducing kernel Hilbert space. Furthermore, we show that each element in H is analytic on Ω if and only if K is analytic on Ω or, equivalently, if and only if 〈K(z),xn〉 is analytic for each n∈N and ||K(·)||H is bounded on all compact subsets of Ω. In this setting, an abstract version of the analytic Kramer theorem is exhibited. Some examples considering different H spaces are given to illustrate these new results.