Notes on ordered reproducing Hilbert spaces over the complex plane

Abstract

Let f , g be entire functions. If there exist M 1 , M 2 > 0 such that |f (z)| M 1 |g(z)| whenever |z| > M 2 we say that f g. Let X be a reproducing Hilbert space with an orthogonal basis {z n } ∞ n=0 . We say that X is an ordered reproducing Hilbert space (or X is ordered) if f g and g ∈ X imply f ∈ X. In this note, we show that if lim inf n→∞ z n+1 / z n = ∞ then X is ordered; if lim inf n→∞ z n+1 / z n = 0 then X is not ordered. In the case lim inf n→∞ z n+1 / z n = l = 0, there are examples to show that X can be of order or opposite.