# Localization measures in the time-scale setting

Localization measures in the time-scale setting Wavelet (or continuous wavelet) transform is superior to the Fourier transform and the windowed (or short-time Fourier) transform because of its ability to measure the time–frequency variations in a signal at different time–frequency resolutions. However, the uncertainty principles in Fourier analysis set a limit to the maximal time–frequency resolution. We present some forms of uncertainty principles for functions that are $$\varepsilon$$ ε -concentrated in a given region within the time–frequency plane involving particularly localization operators. Moreover we show how the eigenfunctions of such localization operators are maximally time–frequency-concentrated in the region of interest and we will use it to approximate such $$\varepsilon$$ ε -concentrated functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Pseudo-Differential Operators and Applications Springer Journals

# Localization measures in the time-scale setting

, Volume 8 (3) – Mar 1, 2017
22 pages

/lp/springer_journal/localization-measures-in-the-time-scale-setting-Qczk2sAsOd
Publisher
Springer International Publishing
Subject
Mathematics; Analysis; Operator Theory; Partial Differential Equations; Functional Analysis; Applications of Mathematics; Algebra
ISSN
1662-9981
eISSN
1662-999X
D.O.I.
10.1007/s11868-017-0195-y
Publisher site
See Article on Publisher Site

### Abstract

Wavelet (or continuous wavelet) transform is superior to the Fourier transform and the windowed (or short-time Fourier) transform because of its ability to measure the time–frequency variations in a signal at different time–frequency resolutions. However, the uncertainty principles in Fourier analysis set a limit to the maximal time–frequency resolution. We present some forms of uncertainty principles for functions that are $$\varepsilon$$ ε -concentrated in a given region within the time–frequency plane involving particularly localization operators. Moreover we show how the eigenfunctions of such localization operators are maximally time–frequency-concentrated in the region of interest and we will use it to approximate such $$\varepsilon$$ ε -concentrated functions.

### Journal

Journal of Pseudo-Differential Operators and ApplicationsSpringer Journals

Published: Mar 1, 2017

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