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A qualitative uncertainty principle and phase retrieval for the Wigner distribution. (Principe d’incertitude qualitatif et reconstruction de phase pour la transformée de Wigner.) (French. Abridged English version) Zbl 0931.42006
For $f,g \in L^2({\Bbb R}^n)$, the Wigner distribution of $f$ and $g$ is defined to be $$ W(f, g)=\int_{{\Bbb R} ^n} f(x+t/2)\overline{g(x-t/2)}\text{ e} ^{2i\pi yt}dt. $$ It is shown that if $f$, $g$ are not identically zero, then the support of $W(f, g)$ has infinite Lebesgue measure. A function $g$ is called a nontrivial Wigner partner of $f$ if $W(f, f)=W(g, g)$ and $g\ne cg$ with $c\in \Bbb C$, $|c|=1$. It is shown that $f$ has a nontrivial Wigner partner if and only if $\text{supp } f$ splits into the union of two disjoint sets $A$ and $B$ of positive Lebesgue measure such that $(A+A)\cup (B+B)$ is disjoint from $A+B$.

42B10Fourier type transforms, several variables
94A12Signal theory (characterization, reconstruction, filtering, etc.)
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