Optimal approximation of discrete-time multirate systems on Hilbert spaces

We study discrete-time (m,n)-multirate systems on separable Hilbert spaces, solving the problem of approximating such a system by one which has a shorter multirate period (m∕q,n∕q), optimally in the Hilbert–Schmidt norm. We work in the state–space setting, providing two state–space representations of the optimal approximant which are expressed in terms of a state–space representation of the original system.