New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform

Abstract

Gabor and wavelet methods are preferred to classical Fourier methods, whenever the time dependence of the analyzed signal is of the same importance as its frequency dependence. However, there exist strict limits to the maximal time-frequency resolution of these both transforms, similar to Heisenberg's uncertainty principle in Fourier analysis. Results of this type are the subject of the following article. Among else, the following will be shown: if $\psi$ is a window function, $f\in L^2(\mathbf{R})\setminus \{0\}$ an arbitrary signal and $G_{\psi }f(\omega ,t)$ the continuous Gabor transform of $f$ with respect to $\psi$, then the support of $G_{\psi }f(\omega ,t)$ considered as a subset of the time-frequency-plane ${\mathbf{R}}^2$ cannot possess finite Lebesgue measure. The proof of this statement, as well as the proof of its wavelet counterpart, relies heavily on the well known fact that the ranges of the continuous transforms are reproducing kernel Hilbert spaces, showing some kind of shift-invariance. The last point prohibits the extension of results of this type to discrete theory.