Prolate Shift Frames and Sampling of Bandlimited Functions

The Shannon sampling theorem can be viewed as a special case of (generalized) sampling reconstructions for bandlimited signals in which the signal is expressed as a superposition of shifts of finitely many bandlimited generators. The coefficients of these expansions can be regarded as generalized samples taken at a Nyquist rate determined by the number of generators and basic shift rate parameter. When the shifts of the generators form a frame for the Paley–Wiener space, the coefficients are inner products with dual frame elements. There is a tradeoff between time localization of the generators and localization of dual generators. The Shannon sampling theorem is an extreme manifestation in which the coefficients are point values but the generating sinc function is poorly localized in time. This work reviews and extends some recent related work of the authors regarding frames for the Paley–Wiener space generated by shifts of prolate spheroidal wave functions, and the question of tradeoff between localization of the generators and of the dual frames is considered.