High-Order Orthonormal Scaling Functions and Wavelets Give Poor Time-Frequency Localization

Chui, Charles K.; Wang, Jianzhong

doi:10.1007/s00041-001-4035-2

Cited by 14 publications

(12 citation statements)

References 4 publications

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“…A continuous orthonormal basis cannot satisfy all of these conditions at the same time. Also, according to Chui and Wang (1995), higher order orthonormal scal-ing functions and wavelets have poor time-scale localisation.…”

Section: Biorthogonal Decompositionmentioning

confidence: 99%

The development of the quaternion wavelet transform

Fletcher

Sangwine

2017

Signal Processing

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The development of the quaternion wavelet transform, Signal Processing, http://dx.doi.org/10. 1016/j.sigpro.2016.12.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. we then go on to detail published approaches to QWTs and to analyse them. We conclude with our own analysis of what it is that should define a QWT as being truly quaternionic and why all but a few of the "QWTs" we have described do not fit our definition.

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Section: Biorthogonal Decompositionmentioning

confidence: 99%

The development of the quaternion wavelet transform

Fletcher

Sangwine

2017

Signal Processing

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“…After all, the original motivation of wavelet decompositions was to localize phase space in the best possible way. Most recently, Chui and Wang have shown that for a special but useful class of wavelets, the standard deviation in phase space increases without bound as the wavelets are made arbitrarily smooth [3,4]. Obviously, such a result cannot hold for wavelets in general, because the Meyer wavelet [5] is a class C ϱ function with finite uncertainty.…”

Section: Introductionmentioning

confidence: 98%

Heisenberg Inequalities for Wavelet States

Battle

1997

Applied and Computational Harmonic Analysis

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We prove a number of uncertainty results for wavelet states, the simplest one being that if a wavelet state is real-valued or, more generally, has zero expected momentum, then the Heisenberg uncertainty is at least 3 2 instead of the universal 1 2 . For wavelet states having a very mild nth-order decay property, we establish a similar result for the uncertainty based on the nth powers of position and momentum, with a lower bound that grows rapidly with n. The proof is extremely elementary. Other Heisenberg inequalities are proven which involve the deviations about the origin of phase space rather than about the mean position of the wavelet in phase space, and the scaling generator plays an even more direct role than in the result mentioned above. The proof is still very elementary, combining the interscale orthogonality property with an iterated application of Rolle's Theorem. Naturally, the lower bounds are much greater for these deviations about the origin of phase space, but this yields consequences for how much ''off-center'' a wavelet must be if its uncertainty is approximately minimized. ᭧ 1997 Academic Press

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“…The smallest possible value of the Heisenberg UC for the family of the Meyer wavelets equals to 6.874 [17]. It is well known [5] that the Heisenberg UC of the Battle-Lemarie and the Daubechies wavelets tends to infinity as their orders grow. A set of real line orthogonal wavelet bases with the uniformly bounded Heisenberg UCs as their orders (smoothness) grow is constructed in [15,16].…”

Section: Introductionmentioning

confidence: 99%