  We characterise asymptotic behaviour of families of symmetric orthonormal
polynomials whose recursion coefficients satisfy certain conditions, satisfied
for example by the (normalised) Hermite polynomials. More generally, these
conditions are satisfied by the recursion coefficients of the form $c(n+1)^p$
for $0<p<1$ and $c>0$, 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 these results to show that, in a
Hilbert space defined in a natural way by such a family of orthonormal
polynomials, every two complex exponentials $e_{\omega}(t)={e}^{i \omega t}$
and $e_{\sigma}(t)={e}^{i \sigma t}$ of distinct frequencies $\omega,\sigma$
are mutually orthogonal. We finally formulate a surprising conjecture for the
corresponding families of non-symmetric orthonormal polynomials; extensive
numerical tests indicate that such a conjecture appears to be true.
