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

Keywords:

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

Abstract

By using a general algebraic identity of Adolf Hurwitz [1], which generalizes an important identity of Abel, we construct a new operator \(S_m^{(\beta_1,\ldots,\beta_m)}\) approximating the functions.
A special case of this is the operator \(Q_m^\beta\) of Cheney-Sharma. We show that this new operator, applied to a function \(f\in C[0,1]\), 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

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

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

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. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2002-vol31-no1-art13

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