ar X iv : 0 71 1 . 48 76 v 1 [ m at h . FA ] 3 0 N ov 2 00 7 Optimal Decompositions of Translations of L 2 - functions

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(R). Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space H, or equivalent the spectral theory of a unitary representation U of the rank-n lattice Zn in Rn. Starting with a non-zero vector ψ ∈ H, we look for relations among the vectors in the cyclic subspace in H generated by ψ. Since these vectors {U(k)ψ|k ∈ Zn} 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-independence. This refers to infinite linear combinations of integral translates of a fixed function with l-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. Mathematics Subject Classification (2000). Primary 47B40, 47B06, 06D22, 62M15; Secondary 42C40, 62M20.