Two Hilbert spaces in which polynomials are not dense

Abstract:

Let $S$ be the Hilbert space of entire functions $f(z)$ such that $||f(z)|{|^2} = \iint {|f(z){|^2}}dm(z)$, where $m$ is a positive measure defined on the Borel sets of the complex plane. Two Hilbert spaces are constructed in which polynomials are not dense. In the second example, our space is one which contains all exponentials and yet in which the exponentials are not complete. This is a somewhat surprising result since the exponentials are always complete on the real line.