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Approximation properties of lagrange interpolation polynomial based on the zeros of (1 −x 2 cosnarccosx

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Abstract

We study some approximation properties of Lagrange interpolation polynomial based on the zeros of (1 − x2)cosnarccosx. By using a decomposition for f(x) ∈ CrCr+1 we obtain an estimate of ‖f(x)−Ln+2(f, x)‖ which reflects the influence of the position of the x's and ω(f(r+1),δ), j = 0, 1, …, s, on the error of approximation.

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References

  1. [1]

    Byrne, G. J., Mills, T. M. and Smith, S. J., On Generalized Hermite-Fejéer Interpolation of Lagrange type on the Chebyshev Nodes, J. Approx. Theory, 105(2000), 263–278.

  2. [2]

    Dzyadyk, V. K., Constructive Characterization of Functions Satisfying the ConditionLipα(0 <α< 1) on a Finite Segment of the Real Axis, Izv. Akad. Nauk SSSR, Ser. Mat., 20(1956), 623–642(in Russian).

  3. [3]

    Gopengauz, I. E., On a Theorem of A.F. Timan on Approximation of Functions by Polynomials on a Finite Interval, Mat. Zametki, 1(1967), 163–172(in Russian).

  4. [4]

    Kis, O., Remarks on the Rapidity of Convergence of Lagrange Interpolation, Ann. Univ. Sci. Budapest, 11(1968), 27–40.

  5. [5]

    Li, X., On the Lagrange Interpolation for a Subset ofC (k) Functions, Constr. Approx., 11(1995), 287–297.

  6. [6]

    Mastroianni, G. and Szabados, J., Jackson Order of Approximation by Lagrange Interpolation, Rend. Circ. Mat. Palermo, 33(1993), 375–386.

  7. [7]

    Mastroianni, G. and Szabados, J., Jackson Order of Approximation by Lagrange Interpolation II, Acta Math. Hungar., 69(1995), 73–82.

  8. [8]

    Szabados, J. and Vértesi, P., Interpolation of Functions, World Scientific. Singapore, 1990.

  9. [9]

    Walsh, J. L., Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Society, Providence, RI, 1969.

  10. [10]

    Xie, T. F. and Zhou, S. P., Approximation of Real Functions, Hangchow Univ. Press, Hangchow, 1998(In Chinese).

  11. [11]

    Zhou, S. P. and Zhu, L. Y., Convergence of Lagrange Interpolation Polynomials for Piecewise Smooth Functions, Acta Math. Hungar., 93(1–2)(2001),71–76.

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Supported by the National Nature Science Foundation.

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Zhu, L. Approximation properties of lagrange interpolation polynomial based on the zeros of (1 −x 2 cosnarccosx . Analysis in Theory and Applications 22, 183–194 (2006). https://doi.org/10.1007/BF03218711

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Key words

  • Lagrange interpolation polynomial
  • zeros of (1 − x2)cosnarccosx
  • piecewise smooth
  • functions
  • error of approximation

AMS(2000) subject classification

  • 41A05
  • 41A25