Asymptotic behaviour of some families of orthonormal polynomials and an associated Hilbert space

We characterise asymptotic behaviour of families of symmetric orthonormal polynomials whose recursion coefficients satisfy certain conditions, satisfied for example by the Hermite polynomials. More generally, these conditions are satisfied by the recursion coefficients of the form $c(n+1)^p$ for $00$, as well as by recursion coefficients which correspond to polynomials orthonormal with respect to the exponential weight $W(x)=\exp(-|x|^\beta)$ for $\beta>1$. We use this result to show that, in a Hilbert space defined in a quite a natural way by such a family of orthonormal polynomials, every two complex exponentials $e_1(t)=exp(i \omega t)$ and $e_2(t)=exp(i \sigma t)$ of distinct positive frequencies $\omega,\sigma$ are mutually orthogonal. Such Hilbert spaces are relevant to signal processing.