On the combinatorial identities of Abel-Hurwitz type and their use in constructive theory of functions.

*(English)*Zbl 1240.41058Summary: 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)].

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 |