This paper provides a generalisation of certain classical Fourier convergence and asymptotic Toeplitz matrix properties to the case where the underlying orthonormal basis is not the conventional trigonometric one, but a rational generalisation which encompasses the trigonometric one as a special case. These generalised Fourier and Toeplitz results have particular application in dynamic system estimation theory. Specifically, the results allow a unified treatment of the accuracy of least squares system estimation using a range of model structures, including those that allow the injection of prior knowledge of system dynamics via the specification of fixed pole or zero locations.