Modeling sampling in tensor products of unitary invariant subspaces

TL;DR

This paper develops a unified sampling theory for tensor products of unitary invariant subspaces in Hilbert spaces, extending existing theories to multiple variables and finite/infinite cases using frame theory.

Contribution

It introduces a comprehensive sampling framework for tensor products of unitary invariant subspaces, merging previous finite and infinite cases and incorporating multiple variables.

Findings

Provides a mathematical framework for sampling in tensor product spaces.

Extends existing theories to multi-variable and infinite-dimensional cases.

Uses frame theory to identify samples as frame coefficients.

Abstract

The use of unitary invariant subspaces of a Hilbert space $\mathcal{H}$ is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of $L^2(\mathbb{R})$ and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows to merge the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature, it also allows to introduce the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both, finite/infinite dimensional cases.