  We investigate a Gaussian quadrature rule and the corresponding orthogonal
polynomials for the oscillatory weight function $e^{i\omega x}$ on the interval
$[-1,1]$. We show that such a rule attains high asymptotic order, in the sense
that the quadrature error quickly decreases as a function of the frequency
$\omega$. However, accuracy is maintained for all values of $\omega$ and in
particular the rule elegantly reduces to the classical Gauss-Legendre rule as
$\omega \to 0$. The construction of such rules is briefly discussed, and though
not all orthogonal polynomials exist, it is demonstrated numerically that rules
with an even number of points are always well defined. We show that these rules
are optimal both in terms of asymptotic order as well as in terms of polynomial
order.
