A class of generalized pseudo-splines

Zhuang, Zhitao; Yang, Jianwei

doi:10.1186/1029-242x-2014-359

Cited by 6 publications

(12 citation statements)

References 8 publications

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“…Proof. The result in (14) clearly holds for all those tuples (z, ), for which 0 < c ≤ |H (γ)| holds for γ in a neighborhood of the origin. Here, the third inequality holds whenever |P (z, ) (γ)| ≥ 1.…”

Section: Lowpass Properties Of Complex Pseudo-splinesmentioning

confidence: 68%

“…If z := m ∈ N in (1) our definition corresponds to the definition of pseudo-spline of type II as given in [3]. In [14,Remark 3] it was claimed that the concept of pseudo-spline can be extended to fractional powers by considering cos r πγ instead of cos 2m πγ, where r ∈ R with r ≥ 2m ∈ N. This, however, is not true since for negative values of the cos-function, the expression cos r πγ = (cos πγ) r is no longer well-defined. For this reason, it is essential to consider H 0 as a function in cos 2 and sin 2 .…”

Section: Pseudo-splines Of Fractional and Complex Ordermentioning

confidence: 99%

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Fractional and complex pseudo-splines and the construction of Parseval frames

Massopust

Forster

Christensen

2017

Applied Mathematics and Computation

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Pseudo-splines of integer order (m, ) were introduced by Daubechies, Han, Ron, and Shen as a family which allows interpolation between the classical B-splines and the Daubechies' scaling functions. The purpose of this paper is to generalize the pseudo-splines to fractional and complex orders (z, ) with α := Re z ≥ 1. This allows increased flexibility in regard to smoothness: instead of working with a discrete family of functions from C m , m ∈ N0, one uses a continuous family of functions belonging to the Hölder spaces C α−1 . The presence of the imaginary part of z allows for direct utilization in complex transform techniques for signal and image analyses. We also show that in analogue to the integer case, the generalized pseudo-splines lead to constructions of Parseval wavelet frames via the unitary extension principle. The regularity and approximation order of this new class of generalized splines is also discussed.MSC Classification (2010): 42C15, 42C40, 65D07

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Section: Lowpass Properties Of Complex Pseudo-splinesmentioning

confidence: 68%

Section: Pseudo-splines Of Fractional and Complex Ordermentioning

confidence: 99%

Fractional and complex pseudo-splines and the construction of Parseval frames

Massopust

Forster

Christensen

2017

Applied Mathematics and Computation

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“…We use the smoothed pseudo-splines introduced in [29] to construct Riesz wavelets and use it to apply our numerical scheme for solving different types of FIDEs. Pseudo-splines of order (p, q) of type I and II, k φ (p,q) , k = 1, 2, are defined in terms of their refinement masks, where…”

Section: Riesz Wavelets Via Smoothed Pseudo-splinesmentioning

confidence: 99%

“…Pseudo-splines known as a generalization of many well-known refinable functions such as the B-splines, interpolate and orthonormal refinable functions [24]. We refer the reader to [22,23,[25][26][27][28][29] and references therein for more details.…”

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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“…More generally, based on the pseudo-splines [7][8][9], Dong and Shen [4] constructed biorthogonal wavelets with prescribed regularity. Zhou and Zheng [5] obtained biorthgonal wavelets from the smoothed pseudo-splines [10].…”

Section: Introductionmentioning

confidence: 99%

Construction of a family of non-stationary biorthogonal wavelets

et al. 2019

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The family of exponential pseudo-splines is the non-stationary counterpart of the pseudo-splines and includes the exponential B-spline functions as special members. Among the family of the exponential pseudo-splines, there also exists the subclass consisting of interpolatory cardinal functions, which can be obtained as the limits of the exponentials reproducing subdivision. In this paper, we mainly focus on this subclass of exponential pseudo-splines and propose their dual refinable functions with explicit form of symbols. Based on this result, we obtain the corresponding biorthogonal wavelets using the non-stationary Multiresolution Analysis (MRA). We verify the stability of the refinable and wavelet functions and show that both of them have exponential vanishing moments, a generalization of the usual vanishing moments. Thus, these refinable and wavelet functions can form a non-stationary generalization of the Coifman biorthogonal wavelet systems constructed using the masks of the D-D interpolatory subdivision.

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A class of generalized pseudo-splines

Abstract: In this paper, a class of refinable functions is given by smoothening pseudo-splines in order to get divergence free and curl free wavelets. The regularity and stability of them are discussed. Based on that, the corresponding Riesz wavelets are constructed.

Cited by 6 publications

References 8 publications

Fractional and complex pseudo-splines and the construction of Parseval frames

Fractional and complex pseudo-splines and the construction of Parseval frames

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

Construction of a family of non-stationary biorthogonal wavelets

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