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

TL;DR

This paper develops sampling theory for $\Lambda$-shift-invariant subspaces of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$, extending classical shift-invariant concepts and motivated by applications in wireless channel estimation.

Contribution

It introduces a framework for sampling in $\Lambda$-shift-invariant subspaces of Hilbert-Schmidt operators, generalizing classical shift-invariant spaces and applying frame theory.

Findings

Sampling results established for these subspaces.

Connections made to channel estimation in wireless communications.

Framework extends classical shift-invariant space theory.

Abstract

The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define $\Lambda$-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice $\Lambda$ in $\mathbb{R}^{2d}$. 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.