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


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


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


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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Hurwitz, A., Über Abel's Vereingemeinerung der Binomischen Formel, Acta Mathematica, 26, pp. 199-203, 1902,

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,

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,




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