The wave front set of the Wigner distribution and instantaneous frequency

Details The wave front set of the Wigner distribution and instantaneous frequency The wave front set of the Wigner distribution and instantaneous frequency

2010-07-06

arxiv.org/abs/1007.0874v1

Mathematics

Functional Analysis

29 pages

Scientific paper

We prove a formula expressing the gradient of the phase function of a function $f: \mathbb R^d \mapsto \mathbb C$ 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 $H^{d/2+1+\epsilon}(\mathbb R^d)$ where $\epsilon>0$. 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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