The wave front set of the Wigner distribution and instantaneous frequency

Abstract

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.

29 pages

References in corpus (1)

• Wave-front sets of Banach function types