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

Elke Wilczok

Documenta mathematica, Tome 5 (2000), pp. 207-226

Résumé

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 ψ is a window function, f∈L2(R)∖{0} an arbitrary signal and Gψ​f(ω,t) the continuous Gabor transform of f with respect to ψ, then the support of Gψ​f(ω,t) considered as a subset of the time-frequency-plane R2 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.

Détail

DOI : 10.4171/dm/79

Classification : 26D10, 43A32, 46C05, 46E22, 81R30, 81S30, 94A12
Mots-clés : wavelets, uncertainty principles, reproducing kernel Hilbert spaces, phase space