A non-stationary combined subdivision scheme generating exponential polynomials

Zheng, Haibin; Zhang, Baoxing

doi:10.1016/j.amc.2017.05.066

Cited by 10 publications

(12 citation statements)

References 16 publications

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“…As is known, the cubic exponential B-spline scheme (13) is 2 -convergent and can generate the function space fl {1, , ± } in the sense of Definition 6 [5]. Thus, it can generate conic sections.…”

Section: Construction Of the Generalized Cubic Exponential B-spline Smentioning

confidence: 99%

“…Siddiqi et al [4] presented ternary nonstationary schemes generating hyperbolas and parabolas. Zheng & Zhang [5] applied the push-back operation in the nonstationary case and constructed a combined nonstationary scheme generating different exponential polynomials. Asghar & Mustafa [6] constructed -ary nonstationary schemes which are new versions of the Lane-Riesenfeld algorithms.…”

Section: Introductionmentioning

confidence: 99%

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A Generalized Cubic Exponential B-Spline Scheme with Shape Control

Zhang

Zheng

Pan

2019

Mathematical Problems in Engineering

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In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters μ and ν. By changing the values of μ and ν, the sensitivity of the shape of the subdivision curve to the initial control value v0 can be changed and different kinds of curves can then be obtained by adjusting the value of v0. For this new scheme, we show that, with any admissible choice of μ and ν, it owns the same smoothness order and support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we construct a similar nonstationary scheme to generate more curves with different shapes and show the role of iterations and suitably chosen functions in the construction and analysis of such schemes. Several examples are given to illustrate the performance of our new schemes.

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Section: Construction Of the Generalized Cubic Exponential B-spline Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Generalized Cubic Exponential B-Spline Scheme with Shape Control

Zhang

Zheng

Pan

2019

Mathematical Problems in Engineering

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“…is section is devoted to the construction of the new family of nonstationary subdivision schemes reproducing exponential polynomials. Before that, we first give an exponential polynomial space, which specializes the space in Lemma 1 and generalizes the space in ( [24], Section 6).…”

Section: Nonstationary Quasi-interpolatory Schemes Reproducing Exponementioning

confidence: 99%

“…For other references on this connection, see also [21][22][23] and the references therein. Yet, unlike the polynomial correction, most of the works related to the push-back operation, except the work in [24], are restricted to the stationary case, and the reproduction property of the obtained interpolatory schemes depends largely on that of the original approximating schemes [20]. In [24], the authors presented a nonstationary combined subdivision generating/reproducing different exponential polynomials using the push-back operation in a suitable way.…”

Section: Introductionmentioning

confidence: 99%

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A Family of Binary Univariate Nonstationary Quasi-Interpolatory Subdivision Reproducing Exponential Polynomials

Zhang

Zheng

Pan

et al. 2019

Mathematical Problems in Engineering

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In this paper, by suitably using the so-called push-back operation, a connection between the approximating and interpolatory subdivision, a new family of nonstationary subdivision schemes is presented. Each scheme of this family is a quasi-interpolatory scheme and reproduces a certain space of exponential polynomials. This new family of schemes unifies and extends quite a number of the existing interpolatory schemes reproducing exponential polynomials and noninterpolatory schemes like the cubic exponential B-spline scheme. For these new schemes, we investigate their convergence, smoothness, and accuracy and show that they can reach higher smoothness orders than the interpolatory schemes with the same reproduction property and better accuracy than the exponential B-spline schemes. Several examples are given to illustrate the performance of these new schemes.

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