Average sampling in certain subspaces of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$

TL;DR

This paper extends average sampling concepts to Hilbert-Schmidt operators on L^2(R^d) using the Weyl transform, establishing new sampling formulas for operator subspaces inspired by shift-invariance.

Contribution

It introduces a novel framework for sampling in operator subspaces via the Weyl transform, bridging phase space functions and Hilbert-Schmidt operators.

Findings

Derived sampling formulas for operator subspaces

Utilized the Weyl transform to connect phase space and operator theory

Extended shift-invariant sampling concepts to Hilbert-Schmidt operators

Abstract

The concept of translation of an operator allows to consider the analogous of shift-invariant subspaces in the class of Hilbert-Schmidt operators. Thus, we extend the concept of average sampling to this new setting, and we obtain the corresponding sampling formulas. The key point here is the use of 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.