# Optimal approximation of discrete-time multirate systems on Hilbert spaces

## TL;DR

This paper addresses the optimal approximation of discrete-time multirate systems on Hilbert spaces by shorter-period systems, providing explicit state-space representations of the optimal solutions in the Hilbert-Schmidt norm.

## Contribution

It introduces a method to approximate multirate systems with shorter periods using state-space representations, advancing the analysis of such systems in Hilbert spaces.

## Key findings

- Explicit state-space formulas for optimal approximants.
- Extension of multirate system approximation theory.
- Application to systems with Hilbert space settings.

## Abstract

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.