Oversampled Wilson expansions

Orthonormal Wilson bases with good time-frequency localization have been constructed by Daubechies, Jaffard, and Journe (1991). We extend this construction to Wilson sets and frames with arbitrary oversampling (or redundancy). We state conditions under which dual Weyl-Heisenberg (WH) sets induce dual Wilson sets, and we formulate duality conditions in the time domain and frequency domain. We show that the dual frame of a Wilson frame has again a Wilson structure, and that it is generated by the dual frame of the underlying Weyl-Heisenberg frame. The Wilson frame construction preserves the numerical properties of the underlying Weyl-Heisenberg frame while halving its redundancy.