Gibbs phenomenon in tight framelet expansions

Mohammad, Mutaz; Lin, En-Bing

doi:10.1016/j.cnsns.2017.06.029

Cited by 20 publications

(16 citation statements)

References 10 publications

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“…We present some numerical illustration by using the dual tight framelets which will generalize the result in Ref. [14]. The results show that if the dual framelet has vanishing moments of order of at least two, then Q n f must exhibit the Gibbs effect.…”

Section: Resultsmentioning

confidence: 61%

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On the Gibbs Effect Based on the Quasi-Affine Dual Tight Framelets System Generated Using the Mixed Oblique Extension Principle

Mohammad

2019

Mathematics

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Gibbs effect represents the non-uniform convergence of the nth Fourier partial sums in approximating functions in the neighborhood of their non-removable discontinuities (jump discontinuities). The overshoots and undershoots cannot be removed by adding more terms in the series. This effect has been studied in the literature for wavelet and framelet expansions. Dual tight framelets have been proven useful in signal processing and many other applications where translation invariance, or the resulting redundancy, is very important. In this paper, we will study this effect using the dual tight framelets system. This system is generated by the mixed oblique extension principle. We investigate the existence of the Gibbs effect in the truncated expansion of a given function by using some dual tight framelets representation. We also give some examples to illustrate the results.

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Section: Resultsmentioning

confidence: 61%

“…[9,10]), wavelets and framelets series (see Refs. [11][12][13][14][15][16][17]), sampling approximations (see Ref. [18]), and many other theoretical investigations (see Refs.…”

Section: Introductionmentioning

confidence: 99%

On the Gibbs Effect Based on the Quasi-Affine Dual Tight Framelets System Generated Using the Mixed Oblique Extension Principle

Mohammad

2019

Mathematics

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“…, m − 1 has a unique solution {c k } m−1 k=1 . In fact, we consider its dual problem: If 20) we must prove that b 1 = · · · = b m−1 = 0. Let q be the unique polynomial such that q ′ = p and q(x 0 ) = 0.…”

Section: )mentioning

confidence: 99%

“…If in addition (2.13) holds and φ is real-valued, then {A n } n∈N must exhibit the Gibbs phenomenon at the origin. [19,20,22,23,25] as special cases on the Gibbs phenomenon of orthogonal wavelets and biorthogonal wavelets. Corollary 3.3 tells us that the truncated approximation using a tight framelet cannot avoid the Gibbs phenomenon while having at least two vanishing moments.…”

Section: Gibbs Phenomenon Of Wavelet and Framelet Expansionsmentioning

confidence: 99%

Gibbs Phenomenon of Framelet Expansions and Quasi-projection Approximation

Han

2019

J Fourier Anal Appl

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The Gibbs phenomenon is widely known for Fourier expansions of periodic functions and refers to the phenomenon that the nth Fourier partial sums overshoot a target function at jump discontinuities in such a way that such overshoots do not die out as n goes to infinity. The Gibbs phenomenon for wavelet expansions using (bi)orthogonal wavelets has been studied in the literature. Framelets (also called wavelet frames) generalize (bi)orthogonal wavelets. Approximation by quasi-projection operators are intrinsically linked to approximation by truncated wavelet and framelet expansions. In this paper we shall establish a key identity for quasi-projection operators and then we use it to study the Gibbs phenomenon of framelet expansions and approximation by general quasiprojection operators. We shall also study and characterize the Gibbs phenomenon at an arbitrary point for approximation by quasi-projection operators. As a consequence, we show that the Gibbs phenomenon appears at all points for every tight or dual framelet having at least two vanishing moments and for quasi-projection operators having at least three accuracy orders. Our results not only improve current results in the literature on the Gibbs phenomenon for (bi)orthogonal wavelet expansions but also are new for framelet expansions and approximation by quasi-projection operators.

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“…This is largely due to the fact that wavelets have the right structure to capture the sparsity in 'physical' images, perfect mathematical properties such as its multi-scale structure, sparsity, smoothness, compactly supported, and high vanish moments. It has many applications in fractional integral and differential equations (see for example [11][12][13][14][15][16][17][18][19][20][21]. Riesz wavelets in L 2 (R) have been used extensively in the context of both pure and numerical analysis in many applications, due to their well prevailing and recognized theory and its natural properties such as sparsity and stability which lead to a well-conditioned scheme.…”

Section: Introductionmentioning

confidence: 99%

The dynamics of COVID-19 in the UAE based on fractional derivative modeling using Riesz wavelets simulation

Mohammad

Trounev

Cattani

2020

Preprint

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The well-known novel virus (COVID-19) is a new strain of coronavirus which considered by the World Health Organization (WHO) as a dangerous epidemic. More than 3.5 million positive cases and 250 thousand deaths (up to May 5, 2020) caused by COVID-19 and has affected more than 280 countries over the world. Therefore, studying the prediction of this virus spreading in further attracts a major public attention. In the United Arab Emirates (UAE), up to same date, there are 14730 positive cases and 137 deaths according to national authorities. In this work, we study a dynamical model based on fractional derivative of nonlinear equations that describe the outbreak of COVID-19 according to the available infection data announced and approved by the national committee in the press. We simulate the available total cases reported based on Riesz wavelets generated by some refinable functions, namely the smoothed pseudo-splines of type I and II with high vanishing moments. Based on these data, we also consider the formulation of the pandemic model using Caputo fractional derivative definition. We then solve numerically the nonlinear system that describe the dynamics of COVID-19 with the given resources based on the collocation Riesz wavelet system constructed. We present graphical illustrations of the numerical solutions of all parameter of the model being handled under different situations. It is anticipated that these results will contribute to the ongoing research to reduce the spreading of the virus and infection cases.

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Gibbs phenomenon in tight framelet expansions

Cited by 20 publications

References 10 publications

On the Gibbs Effect Based on the Quasi-Affine Dual Tight Framelets System Generated Using the Mixed Oblique Extension Principle

On the Gibbs Effect Based on the Quasi-Affine Dual Tight Framelets System Generated Using the Mixed Oblique Extension Principle

Gibbs Phenomenon of Framelet Expansions and Quasi-projection Approximation

The dynamics of COVID-19 in the UAE based on fractional derivative modeling using Riesz wavelets simulation

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