Complex Analysis and Operator Theory

, Volume 2, Issue 3, pp 449–478 | Cite as

Optimal Decompositions of Translations of L 2-Functions

  • Palle E. T. Jorgensen
  • Myung-Sin SongEmail author


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 ψ. 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 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.


Spectrum unitary operators isometries Hilbert space spectral function frames in Hilbert space Parseval frame Riesz Bessel estimates wavelets prediction signal processing 

Mathematics Subject Classification (2000).

Primary 47B40, 47B06, 06D22, 62M15 Secondary 42C40, 62M20 


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Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Department of MathematicsThe University of IowaIowa CityUSA
  2. 2.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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