Optimal Decompositions of Translations of L²-functions

الأصل · EN

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 L²(Rⁿ). 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 L²-independence. This refers to infinite linear combinations of integral translates of a fixed function with l²-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.