Time-frequency localization of multiband signals
The classical "Bell Labs" theory considers spaces of signals that are "essentially" localized to a bounded interval in time and a bounded interval in frequency. Some modern communications applications might benefit from an extension of this theory to cases in which the frequency localization support is a finite union of intervals (multiband). Extensions of the Bell Labs theory would entail estimates of the decay of eigenvalues of compositions of time and frequency localization operators and descriptions of the eigenvectors that are "most localized" in time and frequency. We will pose some of these questions in precise terms and offer some preliminary results relating decay of eigenvalues with the distribution of the frequency support. Numerical approximations of the projections expressed in terms of practical sampling methods and some connections with compressive sampling will also be considered.