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 . 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 then X is ordered; if then X is not ordered. In the case , there are examples to show that X can be of order or opposite.