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On the combinatorial identities of Abel-Hurwitz type and their use in constructive theory of functions. (English) Zbl 1240.41058
Summary: This paper is concerned with the problem of approximation of multivariate functions by means of Abel-Hurwitz-Stancu type linear positive operators. Inspired by the work of D. D. Stancu [Rev. Anal. Numér. Théor. Approx. 31, No. 1, 115–118 (2002; Zbl 1084.41504)], we continue the discussions of the approximation of trivariate functions by a class of Abel-Hurwitz-Stancu operators in the case of trivariate variables, continuous on the unit cube $$K_3=[0,1]^3$$.
In Section 1, which is the Introduction, we give the generalization due to N. H. Abel [J. Reine Angew. Math. 1, 159–160 (1826; Zbl 02751789)] of the Newton binomial formula, and the very important extension of this formula due to A. Hurwitz [Acta Math. 26, 199–204 (1902; JFM 33.0449.04)]. Here also, an interesting combinatorial significance in cycle-free directed graphs is mentioned given by D. E. Knuth [The art of computer programming. Vol. 1: Fundamental algorithms. London: Addison-Wesley Publishing Company (1968; Zbl 0191.17903)]. Then, we present a main result given in 2002 by D. D. Stancu [2002, loc. cit.], where a variant of the Hurwitz identity was employed in order to construct and investigate a new linear positive operator, which is used in the theory of approximation of univariate functions.
In Section 2, we discuss in detail the trivariate polynomial operator of Stancu-Hurwitz type $$S^{(\beta),(\gamma),(\delta)}_{m,n,r}$$ associated to a function $$f\in C(K_3)$$, where $$K_3$$ is the unit cube $$[0,1]^3$$.
Section 3 is devoted to the evaluation of the remainder term of the approximation formula of the function $$f(x,y,z)$$ by means of the Stancu-Hurwitz type operator $$S^{(\beta),(\gamma),(\delta)}_{m,n,r}$$. Firstly, we present an integral form of this remainder, based on the Peano-Milne-Stancu result [D. D. Stancu, J. Soc. Ind. Appl. Math., Ser. B, Numer. Anal. 1, 137–163 (1964; Zbl 0143.07901)]. Then, we give a Cauchy type form for this remainder. By using a theorem of T. Popoviciu, we give an expression, using the divided differences of the first three orders. When the coordinates of the vectors $$(\beta),(\gamma),(\delta)$$ have, respectively, the same values, we are in the case of the second operator of E. W. Cheney and A. Sharma [Riv. Mat. Univ. Parma, II. Ser. 5, 77–84 (1964; Zbl 0146.08202)]. In this case, we obtain an extension of the results from papers by D. D. Stancu [2002, loc. cit.] and by I. Taşcu [“Approximation of bivariate functions by operators of Stancu-Hurwitz type”, Facta Univ., Ser. Math. Inf. 20, 33–39 (2005)].
##### MSC:
 41A35 Approximation by operators (in particular, by integral operators) 41A20 Approximation by rational functions
##### Keywords:
Abel-Hurwitz-Stancu operators