Abstract
We prove a formula expressing the gradient of the phase function of a function f:ℝd↦ℂ as a normalized first frequency moment of the Wigner distribution for fixed time. The formula holds when f is the Fourier transform of a distribution of compact support, or when f belongs to a Sobolev space H d/2+1+ε(ℝd) where ε>0. The restriction of the Wigner distribution to fixed time is well defined provided a certain condition on its wave front set is satisfied. Therefore we first need to study the wave front set of the Wigner distribution of a tempered distribution.
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Communicated by Hans G. Feichtinger.
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Boggiatto, P., Oliaro, A. & Wahlberg, P. The Wave Front Set of the Wigner Distribution and Instantaneous Frequency. J Fourier Anal Appl 18, 410–438 (2012). https://doi.org/10.1007/s00041-011-9201-6
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DOI: https://doi.org/10.1007/s00041-011-9201-6