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

By Elke Wilczok. Documenta Mathematica, 2000

149 citations recorded by the source corpus.

Gabor and wavelet methods are preferred to classical Fourier methods, whenever the time dependence of the analyzed sig- nal 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 is a window function, f 2 L 2 (R) n f0g an arbitrary signal and G f(!;t) the con- tinuous Gabor transform of f with respect to , then the support of G f(!;t) considered as a subset of the time-frequency-plane R 2 can- not possess nite 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 reproduc- ing kernel Hilbert spaces, showing some kind of shift-invariance. The last point prohibits the extension of results of this type to discrete theory.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/ce4826fc11d93be713d441d0a53becc5d31c2c53

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Saifallah Ghobber
2. Slim Omri
3. Hatem Mejjaoli
4. Nadia Sraieb
5. Philippe Jaming

In Atomic and Molecular Physics, and Optics

1. Saifallah Ghobber
2. Slim Omri
3. Hatem Mejjaoli
4. Nadia Sraieb

In Computer Vision and Pattern Recognition

1. Slim Omri
2. Hatem Mejjaoli
3. Nadia Sraieb
4. Saifallah Ghobber

Citations in the corpus

• Time–frequency concentration of the windowed Hankel transform (2014, methodology): “10) The proofs in [12,13] are based on Benedicks’ uncertainty principle [10] and the proofs in [5,7] are based on Amrein-Berthier’s techniques [9] and their uncertainty inequality.”
• Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms (2001, unknown): “Wilczock [29].”
• Uncertainty Principles for the Continuous Dunkl Gabor Transform and the Dunkl Continuous Wavelet Transform (2008, background): “Recently, dealing with these problems of uncertainty principles, a number of papers has been published (see [7],[11] for Dunkl transform, [1],[ 19 ],[20] for Gabor and wavelet transform, [9] for hypergroups and others ... ).”