Sampling in Shift-Invariant-Like Subspaces of Hilbert-Schmidt Operators

The concept of translation of an operator, defined by using conjugation with time-frequency shifts, allows to consider the analogue of shift-invariant subspaces in the class of Hilbert-Schmidt operators. Thus, one can define \(\Lambda \) -shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice \(\Lambda \) in \(\mathbb {R}^{2d}\) , and extend the concept of average sampling to this new setting. The key point here is the use of the Konh-Nirenberg (or the Weyl) transform, a unitary mapping between the space of square integrable functions in the phase space \(\mathbb {R}^d\times \widehat {\mathbb {R}}^d\) and the Hilbert space of Hilbert-Schmidt operators on \(L^2(\mathbb {R}^d)\) , which permits to take advantage of some well established sampling results. Obtaining sampling results for these subspaces appears as a natural question that can be motivated by the problem of channel estimation in wireless communications. These sampling results are obtained in the light of the frame theory in a separable Hilbert space.