Notes on ordered reproducing Hilbert spaces over the complex plane

Let f , g be entire functions. If there exist M1,M2 > 0 such that |f (z)| M1|g(z)| whenever |z|>M2 we say that f g. Let X be a reproducing Hilbert space with an orthogonal basis {zn}∞ 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 infn→∞ zn+1 / zn =∞then X is ordered; if lim infn→∞ zn+1 / zn =0 then X is not ordered. In the case lim infn→∞ zn+1 / zn = l = 0, there are examples to show that X can be of order or opposite. © 2005 Elsevier Inc. All rights reserved