2007/11/30 by Palle E. T. Jorgensen, Jorgensen, Palle E. T., Myung-Sin Song +1
Mathematics · #06D22 #42C40 #47B06 #47B40 #62M15 #62M20 #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP) #math.FA #math.SP #msc:06D22 #msc:42C40 #msc:47B06 #msc:47B40 #msc:62M15 #msc:62M20
paper · pdf · doi:10.48550/arxiv.0711.4876
30 pages, 3 figures
arxiv created 2007/11/30 · arxiv updated 2009/12/01
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.