toeplitz operators are fundamental and ubiquitous in signal processing and information theory as models for convolutional (filtering) systems. due to the fact that any practical system can access only signals of finite duration, however, time-limited restrictions of toeplitz operators are also of interest. in the discrete-time case, time-limited toeplitz operators are simply toeplitz matrices. in this talk we survey existing and present new bounds on the eigenvalues (spectra) of time-limited toeplitz operators, and we discuss applications of these results in various signal processing contexts. as a special case, we discuss time-frequency limiting operators, which alternatingly limit a signal in the time and frequency domains. slepian functions arise as eigenfunctions of these operators, and we describe applications of slepian functions in spectral analysis of multiband signals, super-resolution sar imaging, and blind beamforming in antenna arrays. \\ \\ this talk draws from joint work with numerous collaborators including zhihui zhu from the university of denver.