The wave front set of the Wigner distribution and instantaneous frequency

2010/07/06 by Boggiatto, Paolo, Oliaro, Alessandro, Wahlberg, Patrik
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1007.0874

Abstract

We prove a formula expressing the gradient of the phase function of a function f: \mathbb R^d ↦ \mathbb 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(\mathbb R^d) 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 study the wave front set of the Wigner distribution of a tempered distribution.

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