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What is the Wigner function closest to a given square integrable function?
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Abstract(s)
We consider an arbitrary square integrable function F on the phase-space and look for the Wigner function closest to it with respect to the L2 norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert--Schmidt operator withWeyl symbol F. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a bona fide Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results, we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics.
Description
Keywords
Weyl quantization Wigner functions Wigner approximation Hilbert--Schmidt operators
Citation
Ben-Benjamin, J. S., Cohen, L., Dias, N. C., Loughlin, P., & Prata, J. N. (2018). What is the Wigner function closest to a given square integrable function?. SIAM Journal on Mathematical Analysis, 50(5), 5161-5197.