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Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation

Published online by Cambridge University Press:  28 May 2015

Guiqiao Xu  and
Zheng Zhang
Guiqiao Xu*
Affiliation:
Department of Mathematics, Tianjin Normal University, Tianjin, 300387, P.R. China
Zheng Zhang
Affiliation:
Department of Mathematics, Tianjin Normal University, Tianjin, 300387, P.R. China
*
*Corresponding author.Email address:Xuguiqiao@eyou.com
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Abstract

For the approximation in Lp-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

41A1565D0741A28Piecewise cubic Hermite interpolationLp-normsimultaneous approximationequidistant knotinfinite-dimensional Kolmogorov width
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 3 , August 2014 , pp. 317 - 333
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Babenko, V. F., On the best uniform appriximation by splines with restrictions imposed on their derivatives, Mat. Zametki, vol. 50, no. 6 (1991), pp. 24–29.Google Scholar
[2]Babenko, V. F., On the best L1-appriximation by splines with restrictions imposed on their derivatives, Mat. Zametki, vol. 51, no. 5 (1992), pp. 12–19.Google Scholar
[3]Babenko, V. F., The best L1 approximation of the classes W1r by splines from W1r, Ukr. Mat. Zh., vol. 46, no. 10 (1994), pp. 1410–1413.CrossRefGoogle Scholar
[4]Babenko, V. F., Parfinovich, N. V., On the best L1 approximations of function classes by splines under restrictions imposed on their derivatives, Ukr. Mat. Zh., vol. 51, no. 4 (1999), pp. 481–491.CrossRefGoogle Scholar
[5]Bergh, J., Lofstrom, J., Interpolation spaces, An Introduction, Springer-Verlag Berlin Heidelberg, 1976.CrossRefGoogle Scholar
[6]Kigore, T., An elementary simultaneous approximation theorem, Proc. Amer. Math. Soc., vol. 118 (1993), pp. 529–536.Google Scholar
[7]Konovalov, V. N., Approximation of Sobolev classes by their finite-dimensional sections, Mat. Zametki, vol. 72, no. 3 (2002), pp. 370–382.Google Scholar
[8]Konovalov, V. N., Approximation of Sobolev classes by their sections of finite dimension, Ukr. Mat. Zh., vol. 54, no, 5 (2002), pp. 795–805.CrossRefGoogle Scholar
[9]Korneichuk, N. P., Splines in the theory of approximations, Nauka, Moscow, 1984 (in Russian).Google Scholar
[10]Li, C., Infinite-dimensional widths in the space of functions(II), J. Approx. Theory, vol. 69 (1992), pp. 15–34.CrossRefGoogle Scholar
[11]Szabados, J., Vestesi, P., A survey on mean convergence of interpolatory processes, J. Comput. Appl. Math., vol. 43 (1992), pp. 3–18.CrossRefGoogle Scholar
[12]Trigub, R., Approximation of functions by polynomials with integral coefficients, Izv. Akad. Nauk. SSSR Ser. Mat., vol. 26 (1962), pp. 261–280 (in Russian).Google Scholar
[13]Vertesi, P., Xu, Y., Mean convergence of Hermite interpolation revisited, Acta Math. Hungar, vol. 69, no. 2 (1995), pp. 185–210.CrossRefGoogle Scholar
[14]Xiao, W. W., Relative infinite-dimensional widths of Sobolev classes Wpr(R), J. Math. Anal. Appl., vol. 369, no. 2 (2010), pp. 575–582.CrossRefGoogle Scholar
[15]Xu, G. Q., The relative n-widths of Sobolev classes with restrictions, J. Approx. Theory, vol. 157 (2009), pp. 19–31.CrossRefGoogle Scholar