We consider problems related to the generalization of the classical Fourier basis to a basis of rational functions with prescribed poles outside the unit disk. We give some generalizations about the convergence and estimation of the Fourier coefficients with respect to this generalized basis. We also consider a rational generalization of the classical Toeplitz matrices and consider the asymptotic distribution of their spectrum. These bases and general Toeplitz matrices were considered by Ninness et al. in the context of least squares system estimation where the prescribed poles allow to incorporate a priori knowledge into the system dynamics of the model.