  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.
