Abstract
We study Daubechies’ time–frequency localization operator, which is characterized by a window and weight function. We consider a Gaussian window and a spherically symmetric weight as this choice yields explicit formulas for the eigenvalues, with the Hermite functions as the associated eigenfunctions. Inspired by the fractal uncertainty principle in the separate time–frequency representation, we define the n-iterate mid-third spherically symmetric Cantor set in the joint representation. For the n-iterate Cantor set, precise asymptotic estimates for the operator norm are then derived up to a multiplicative constant.
Highlights
The problem of localizing signals in time and frequency is an old and important one in signal analysis
Under the above condition, let Pn denote the Daubechies operator localizing on the n-iterate spherically symmetric Cantor set defined in the disk of radius R > 0
If we interpret f as an amplitude signal depending on time, its Fourier transform fcorresponds to a frequency representation of the signal
Summary
The problem of localizing signals in time and frequency is an old and important one in signal analysis. We often wish to analyze signals on different time– frequency domains, and we would attempt to concentrate signals on said domains. Different approaches for how to construct such time–frequency localization operators have been suggested, either based on a separate or joint time–frequency representation of the signal (see [4,13]). The localization operators, regardless of Communicated by Elena Cordero. Engineering, NTNU – Norwegian University of Science and Technology, 7034 Trondheim, Norway
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