Optimal Decompositions of Translations of $L^{2}$-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 $ψ\in \mathcal{H}$, we look for relations among the vectors in the cyclic subspace in $\mathcal{H}$ generated by $ψ$. Since these vectors $\{U(k)ψ| k \in \mathbb{Z}^{n}\}$ 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.