  The aim of this article is to present a time-frequency theory for orthogonal
polynomials on the interval [-1,1] that runs parallel to the time-frequency
analysis of bandlimited functions developed by Landau, Pollak and Slepian. For
this purpose, the spectral decomposition of a particular compact
time-frequency-operator is studied. This decomposition and its eigenvalues are
closely related to the theory of orthogonal polynomials. Results from both
theories, the theory of orthogonal polynomials and the Landau-Pollak-Slepian
theory, can be used to prove localization and approximation properties of the
corresponding eigenfunctions. Finally, an uncertainty principle is proven that
reflects the limitation of coupled time and frequency locatability.

Classical Analysis and ODEs