New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform (Q1978055)

Given a window function \(\psi \in L^2(\mathbb R)\), the Gabor transform of a function \(f\in L^2(\mathbb R)\) is given by \(Gf(\omega,t)= \frac{1}{\sqrt{2\pi}} \int_\infty^\infty f(x) \overline{\psi(x-t)}e^{-i\omega x} dx\). It is proved that if \(f\neq 0\), then the support of \(Gf\) has infinite Lebesgue measure. A similar result is stated for the wavelet transform. An abstract framework including both cases is presented.