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New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. (English) Zbl 0947.42024

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.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
43A32 Other transforms and operators of Fourier type
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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