Sampling in $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators on $ L^2(\mathbb{R}^d) $

Abstract

<p style='text-indent:20px;'>The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define <inline-formula><tex-math id="M3">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula>-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice <inline-formula><tex-math id="M4">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{R}^{2d} $\end{document}</tex-math></inline-formula>. These spaces can be seen as a generalization of classical shift-invariant subspaces of square integrable functions. 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.</p>