  Toeplitz operators are fundamental and ubiquitous in signal processing and
information theory as models for linear, time-invariant (LTI) systems. Due to
the fact that any practical system can access only signals of finite duration,
time-limited restrictions of Toeplitz operators are naturally of interest. To
provide a unifying treatment of such systems working on different signal
domains, we consider time-limited Toeplitz operators on locally compact abelian
groups with the aid of the Fourier transform on these groups. In particular, we
survey existing results concerning the relationship between the spectrum of a
time-limited Toeplitz operator and the spectrum of the corresponding
non-time-limited Toeplitz operator. We also develop new results specifically
concerning the eigenvalues of time-frequency limiting operators on locally
compact abelian groups. Applications of our unifying treatment are discussed in
relation to channel capacity and in relation to representation and
approximation of signals.
