Combined Shepard operators with Chebyshev nodes

Authors

  • Cristina O. Oşan “Babes-Bolyai” University, Cluj-Napoca, Romania
  • Radu T. Trîmbitaş “Babes-Bolyai” University, Cluj-Napoca, Romania

DOI:

https://doi.org/10.33993/jnaat331-761

Keywords:

Shepard interpolation, Chebyshev nodes
Abstract views: 197

Abstract

In this paper we study combined Shepard-Lagrange univariate interpolation operator\[S_{n,\mu}^{L,m}(Y;f,x):=S_{n,\mu}^{L,m}(f,x)=\frac{\sum\limits_{k=0}^{n+1}\left\vert x-y_{n,k}\right\vert ^{-\mu}(L_{m}f)(x,y_{n,k})}{\sum\limits_{k=0}^{n+1}\left\vert x-y_{n,k}\right\vert ^{-\mu}},\]where \((y_{n,k})\) are the interpolation nodes and \((L_{m}f)(x;y_{n,k})\) is the Lagrange interpolation polynomial with nodes \( y_{n,k},\, y_{n,k+1}, \, y_{n,k+2}, \ldots, \, y_{n,k+m} \), when the interpolation nodes \((y_{n,k})_{k=\overline{1,n}}\) are the zeros of first kind Chebyshev polynomial completed with \(y_{n,0}=-1\)and \(y_{n,n+1}=1\). We give a direct proof for error estimation and some numerical examples.

Downloads

Download data is not yet available.

References

Gh. Coman and R. Trîmbiţaş, Combined Shepard univariate operators, East Journal on Approximations, 7, no. 4, pp. 471-483, 2001.

G. Criscuolo and G. Mastroianni, Estimates of the Shepard interpolatory procedure, Acta. Math. Hung., 61, nos. 1-2, pp. 79-91, 1993, https://doi.org/10.1007/bf01872100 DOI: https://doi.org/10.1007/BF01872100

M. Crouzeix and A. L. Mignot, Analyse numérique des équations différentielles, 2e édition, Masson, Paris, 1989.

B. Della Vecchia and G. Mastroianni, Pointwise estimates of rational operators based on general distribution of knots, Facta Universitatis (Niš), Ser. Math. Inform., 6, pp. 63-78, 1991.

B. Della Vecchia and G. Mastroianni, Pointwise simultaneous approximation by rational operators, J. Approx. Theory, 65, pp. 140-150, 1991, https://doi.org/10.1016/0021-9045(91)90099-v DOI: https://doi.org/10.1016/0021-9045(91)90099-V

R. Trîmbiţaş, Univariate Shepard-Lagrange interpolation, Kragujevac J. Math., 24, pp. 85-94, 2002.

D. Shepard, A two dimensional interpolation function for irregularly spaced data, Proc. 23rd Nat. Conf. ACM, pp. 517-523, 1968. DOI: https://doi.org/10.1145/800186.810616

Downloads

Published

2004-02-01

How to Cite

Oşan, C. O., & Trîmbitaş, R. T. (2004). Combined Shepard operators with Chebyshev nodes. Rev. Anal. Numér. Théor. Approx., 33(1), 73–78. https://doi.org/10.33993/jnaat331-761

Issue

Section

Articles