Optimal Decompositions of Translations of $L^{2}$-functions

TL;DR

This paper develops a computational method to analyze spectral functions of commuting operators, with applications in wavelets, signal processing, and spectral theory, focusing on translation invariance and linear relations in Hilbert spaces.

Contribution

It introduces a general approach to spectral density analysis for families of commuting unitary operators, extending to stochastic integrals and providing geometric characterizations of linear relations.

Findings

Spectral densities for commuting unitary operators are characterized.

The approach applies to wavelet and frame theory contexts.

Results include geometric descriptions of linear relations among translated functions.

Abstract

In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space $L^{2}(\mathbb{R}^{n})$. Our approach applies more generally to families of $n$ arbitrary commuting unitary operators in a complex Hilbert space $\mathcal{H}$, or equivalent the spectral theory of a unitary representation $U$ of the rank-$n$ lattice $\mathbb{Z}^{n}$ in $\mathbb{R}^{n}$. Starting with a non-zero vector $\psi \in \mathcal{H}$, we look for relations among the vectors in the cyclic subspace in $\mathcal{H}$ generated by $\psi$. Since these vectors $\{U(k)\psi | k \in \mathbb{Z}^{n}\}$ involve infinite ``linear combinations," the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name $L^{2}$-independence. This refers to \textit{infinite} linear combinations of integral translates of a fixed function with $l^{2}$-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals.