Use of identity of A. Hurwitz for construction of a linear positive operator of approximation

Authors

  • Dimitrie D. Stancu “Babes-Bolyai” University, Cluj-Napoca, Romania

DOI:

https://doi.org/10.33993/jnaat311-714

Keywords:

Hurwitz's identity, Abel's generalization of the binomial formula, linear positive operator of approximation, the Peano theorem, divided difference
Abstract views: 315

Abstract

By using a general algebraic identity of Adolf Hurwitz [1], which generalizes an important identity of Abel, we construct a new operator Sm(β1,…,βm) approximating the functions.
A special case of this is the operator Qmβ of Cheney-Sharma. We show that this new operator, applied to a function f∈C[0,1], is interpolatory at both sides of the interval [0,1], and reproduces the linear functions. We also give an integral representation of the remainder of the approximation formula of the function f by means of this operator. By applying a criterion of T. Popoviciu [2], is also given an expression of this remainder by means of divided difference of second order.

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References

Hurwitz, A., Über Abel's Vereingemeinerung der Binomischen Formel, Acta Mathematica, 26, pp. 199-203, 1902, https://doi.org/10.1007/BF02415491 DOI: https://doi.org/10.1007/BF02415491

Popoviciu, T., Sur le reste dans certaines formules linéaires d'approximation de l'analyse, Mathematica (Cluj), 1 (24), pp. 95-142, 1959,

Stancu, D. D., Evaluation of the remainder term in approximation formulas by Bernstein polynomials, Math. Comp., 17, pp. 270-278, 1963, https://doi.org/10.1090/S0025-5718-1963-0179524-6 DOI: https://doi.org/10.1090/S0025-5718-1963-0179524-6

Stancu, D. D. and Cismaşiu, C., On an approximating linear positive operator of Cheney-Sharma, Rev. Anal. Numér. Théor. Approx., 26, pp. 221-227, 1997, http://ictp.acad.ro/jnaat/journal/article/view/1997-vol26-nos1-2-art30

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Published

2002-02-01

Issue

Vol. 31 No. 1 (2002)

Section

Articles

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Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory

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Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

How to Cite

Stancu, D. D. (2002). Use of identity of A. Hurwitz for construction of a linear positive operator of approximation. Rev. Anal. Numér. Théor. Approx., 31(1), 115-118. https://doi.org/10.33993/jnaat311-714