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.