Symmetric Wilson systems

Abstract

The Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé using combinations of elements of Gabor tight frame with redundancy 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Then, Kutyniok and Strohmer generalized the notion of Wilson system to the lattices whose generator matrix is in Hermite normal form.In the present publication, we introduce Wilson systems where the time-frequency shifts are combined symmetrically with respect to the origin. Using the arguments from previous papers that work also in this case, we give a full characterization of atoms for which the obtained Wilson systems are orthonormal bases in [Formula: see text] and we generalize these results to different sign sequences and to symplectic lattices of density [Formula: see text].