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Volume 7, Issue 3
Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation

Guiqiao Xu & Zheng Zhang

DOI: 10.4208/nmtma.2014.1232nm

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 317-333.

Published online: 2014-07

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  • Abstract

For the approximation in $L_p$-norm, we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots. For $p = 1$, $∞$, we obtain its values. By these results we know that for the Sobolev classes, the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths. At the same time, the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.

  • Keywords

Piecewise cubic Hermite interpolation, $L_p$-norm, simultaneous approximation, equidistant knot, infinite-dimensional Kolmogorov width.

  • AMS Subject Headings

41A15, 65D07, 41A28

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-7-317, author = {Guiqiao Xu and Zheng Zhang}, title = {Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {3}, pages = {317--333}, abstract = {

For the approximation in $L_p$-norm, we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots. For $p = 1$, $∞$, we obtain its values. By these results we know that for the Sobolev classes, the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths. At the same time, the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.1232nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5877.html} }
TY - JOUR T1 - Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation AU - Guiqiao Xu & Zheng Zhang JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 317 EP - 333 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.1232nm UR - https://global-sci.org/intro/article_detail/nmtma/5877.html KW - Piecewise cubic Hermite interpolation, $L_p$-norm, simultaneous approximation, equidistant knot, infinite-dimensional Kolmogorov width. AB -

For the approximation in $L_p$-norm, we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots. For $p = 1$, $∞$, we obtain its values. By these results we know that for the Sobolev classes, the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths. At the same time, the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.

Guiqiao Xu and Zheng Zhang. (2014). Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation. Numerical Mathematics: Theory, Methods and Applications. 7 (3). 317-333. doi:10.4208/nmtma.2014.1232nm
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