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Volume 7, Issue 1
On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline $S_3^{1,2}(∆_{mn}^{(2)})$

Jiang Qian & Fan Wang

DOI: 10.4208/nmtma.2014.y12035

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 1-22.

Published online: 2014-07

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  • Abstract

In this paper, based on the basis composed of two sets of splines with distinct local supports, cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation. The variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points, which can reproduce any polynomial of nearly best degrees. And by means of the modulus of continuity, the estimation of the operator approximating a real sufficiently smooth function is reviewed as well. Moreover, the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation. And then the convergence results are worked out.

  • Keywords

Bivariate splines, conformality of smoothing cofactor method, nonuniform type-2 triangulation, quasi-interpolation, modulus of continuity.

  • AMS Subject Headings

65D07, 41A25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-7-1, author = {Jiang Qian and Fan Wang}, title = {On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline $S_3^{1,2}(∆_{mn}^{(2)})$}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {1}, pages = {1--22}, abstract = {

In this paper, based on the basis composed of two sets of splines with distinct local supports, cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation. The variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points, which can reproduce any polynomial of nearly best degrees. And by means of the modulus of continuity, the estimation of the operator approximating a real sufficiently smooth function is reviewed as well. Moreover, the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation. And then the convergence results are worked out.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.y12035}, url = {http://global-sci.org/intro/article_detail/nmtma/5863.html} }
TY - JOUR T1 - On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline $S_3^{1,2}(∆_{mn}^{(2)})$ AU - Jiang Qian & Fan Wang JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 1 EP - 22 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.y12035 UR - https://global-sci.org/intro/article_detail/nmtma/5863.html KW - Bivariate splines, conformality of smoothing cofactor method, nonuniform type-2 triangulation, quasi-interpolation, modulus of continuity. AB -

In this paper, based on the basis composed of two sets of splines with distinct local supports, cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation. The variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points, which can reproduce any polynomial of nearly best degrees. And by means of the modulus of continuity, the estimation of the operator approximating a real sufficiently smooth function is reviewed as well. Moreover, the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation. And then the convergence results are worked out.

Jiang Qian and Fan Wang. (2014). On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline $S_3^{1,2}(∆_{mn}^{(2)})$. Numerical Mathematics: Theory, Methods and Applications. 7 (1). 1-22. doi:10.4208/nmtma.2014.y12035
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