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Simultaneouse approximation to a differentiable function and its derivatives by Lagrange interpolating polynomials

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Approximation Theory and its Applications

Abstract

This paper establishes the following pointwise result for simultaneous Lagrange interpolating approximation: Let f∈C q[−1,1] and r=[q+2/2], then

$$\left| {f^{(k)} (x) - P_n^{(k)} (f,x)} \right| = O(1)\Delta _n^{q - k} (x)\omega (f^{(q)} ,\Delta _n (x))(||L_n || + ||L_n $$

where Pn(f,x) is the Lagrange interpolating polynomial of degree n+2r...1 of f(x) on the nodes Xn∪Yn (see the definition of the next),\(\Delta _n (x) = \frac{{N1 - x2}}{n} + \frac{1}{{n^2 }}\).

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Authors and Affiliations

  1. China Insititue of Metrology, 310034, Hangzhou, P. R. China

    T. F. Xie

  2. Department of Mathematics, University of Alberta, T6G 2G1, Edmonton, Alberta, Canada

    S. P. Zhou

Authors
  1. T. F. Xie
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  2. S. P. Zhou
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Additional information

The second named author was supported in part by an NSERC Postdoctoral Fellowship, Canada and a CRF Grant, University of Alberta

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Xie, T.F., Zhou, S.P. Simultaneouse approximation to a differentiable function and its derivatives by Lagrange interpolating polynomials. Approx. Theory & its Appl. 10, 100–109 (1994). https://doi.org/10.1007/BF02837043

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  • DOI: https://doi.org/10.1007/BF02837043

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