A kind of Bernoulli-type quasi-interpolation operator with univariate multiquadrics

Wang, Ren-Hong; Xu, Min

doi:10.1590/s1807-03022010000100004

Comput. Appl. Math.

2010

DOI: 10.1590/s1807-03022010000100004

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A kind of Bernoulli-type quasi-interpolation operator with univariate multiquadrics

Ren-Hong Wang¹,

Min Xu²

Abstract: Abstract.In this paper, a kind of Bernoulli-type operator is proposed by combining a univariate multiquadric quasi-interpolation operator with the generalized Taylor polynomial. With an assumption on the shape-preserving parameter c, the convergence rate of the new operator is derived, which indicates that it could produce the desired precision. Numerical comparisons show that this method offers a higher degree of accuracy. Moreover, the associated algorithm is easily implemented.Mathematical subject classific… Show more

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2023

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Cited by 3 publications

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A kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator with higher approximation order

2023

J Inequal Appl

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In this paper, a kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator is studied by combining the known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate Bernoulli polynomials. Some error bounds and convergence rates of the combined operators are studied. A selection of numerical examples is presented to compare the performances of the obtained scheme. Furthermore, our method can be applied to time-dependent differential equations. Its advantage is that the algorithm is very simple and easy to implement.

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A kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator with higher approximation order

2023

J Inequal Appl

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Bivariate multiquadric quasi-interpolation operators of Lidstone type

2023

MATH

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<abstract><p>In this paper, a kind of bivariate multiquadric quasi-interpolant with the derivatives of a approximated function is studied by combining the known multiquadric quasi-interpolant with the generalized Taylor polynomials that act as the bivariate Lidstone interpolation polynomials. For practical purposes, a kind of improved approximation operator without any derivative of the approximated function is given by using bivariate divided differences to approximate the derivatives. It has the property of high-degree polynomial reproducing. In addition, the improved bivariate quasi-interpolation operators only demand information of the location points rather than the derivatives of the function approximated. Some error bounds in terms of the modulus of continuity of high order and Peano representations for the error are given. Several numerical comparisons with other existing methods are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the algorithm is very simple and easy to implement.</p></abstract>

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A kind of even order Bernoulli-type operator with bivariate Shepard

2023

MATH

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<abstract><p>It is known that an efficient method for interpolation of very large scattered data sets is the method of Shepard. Unfortunately, it reproduces only the constants. In this paper, we first generalize an expansion in bivariate even order Bernoulli polynomials for real functions possessing a sufficient number of derivatives. Finally, by combining the known Shepard operator with the even order Bernoulli bivariate operator, we construct a kind of new approximated operator satisfying the higher order polynomial reproducibility. We study this combined operator and give some error bounds in terms of the modulus of continuity of high order and also with Peano's theorem. Numerical comparisons show that this new technique provides the higher degree of accuracy. Furthermore, the advantage of our method is that the algorithm is very simple and easy to implement.</p></abstract>

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A kind of Bernoulli-type quasi-interpolation operator with univariate multiquadrics

Cited by 3 publications

References 15 publications

A kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator with higher approximation order

A kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator with higher approximation order

Bivariate multiquadric quasi-interpolation operators of Lidstone type

A kind of even order Bernoulli-type operator with bivariate Shepard

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