Iterates of multidimensional approximation operators via Perov theorem


The starting point is an approximation process consisting of linear and positive operators. The purpose of this note is to establish the limit of the iterates of some multidimensional approximation operators. The main tool is a Perov’s result which represents a generalization of Banach fixed point theorem. In order to support the theoretical aspects, we present three applications targeting respectively the operators Bernstein, Cheney-Sharma and those of binomial type. The last class involves an incursion into umbral calculus.


Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Radu Precup
Institute of Advanced Studies in Science and Technology, Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy


Linear positive operator, contraction principle, Perov theorem, Bernstein operator, CheneySharma operator, umbral calculus, binomial operator.

Paper coordinates

O. Agratini, R. Precup, Iterates of multidimensional approximation operators via Perov theorem, Carpathian J. Math., 38 (2022) no. 3, pp. 539-546,


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Carpathian Journal Math.

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1584 – 2851

Online ISSN

1843 – 4401

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[1] Agratini, O., Rus, I. A. Iterates of a class of discrete linear operators via contraction principle. Comment. Math. Univ. Carolinae 44 (2003), 555–563.

[2] Agratini, O., Rus, I.A. Iterates of some bivariate approximation process via weakly Picard operators. Nonlinear Anal. Forum 8 (2003), no. 2, 159–168.

[3] Berinde, V., Approximating fixed points of weak contractions using Picard iteration. Nonlinear Anal. Forum 9 (2004), no. 1, 43–53.

[4] Catinas, T., Iterates of a modified Bernstein type operator. J. Numer. Anal. Approx. Theory 48 (2019), no. 2, 144–147.

[5] Cheney, E. W., Sharma, A., On a generalization of Bernstein polynomials. Riv. Mat. Univ. Parma (2) 5 (1964), 77–84.

[6] Craciun, M., Approximation operators constructed by means of Sheffer sequences. Rev. Anal. Numer. Theor. Approx. 30 (2001), no. 2, 135–150.

[7] Cvetkovic, M., Rakocevic, V., Extensions of Perov theorem. Carpathian J. Math. 31 (2015), no. 2, 181–188.

[8] Filip, A.-D., Petrusel, A,. Existence and uniqueness of the solution for a general system of Fredholm integral equations. Math. Methods Appl. Sci. (2020),

[9] Gavrea, I., Ivan, M., On the iterates of positive linear operators preserving the affine functions. J. Math. Anal. Appl. 372 (2010), 366–368.

[10] Jensen, J. L., W. V. Sur une identite d’Abel et sur d’autres formules analogues. Acta Math. 26 (1902), 307–318.

[11] Kelisky, R. P., Rivlin, T. J. Iterates of Bernstein operators. Pacific J. Math. 21 (1967), 511–520.

[12] Novac, A., Precup, R. Perov type results in gauge spaces and their applications to integral systems on semi-axis. Math. Slovaca 64 (2014), 961–972.

[13] Perov, A. I., On Cauchy problem for a system of ordinary differential equation. Priblizhen. Metody Reshen. Differ. Uravn. 2 (1964), 115–134.

[14] Popoviciu, T., Remarques sur le polynomes binomiaux. Bul. Soc. Sci. Cluj (Roumanie) 6 (1931), 146–148 (also reproduced in Mathematica (Cluj) 6 (1932), 8–10).

[15] Precup, R., The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Modelling 49 (2009), 703–708.

[16] Rota, G.-C., Kahaner, D.; Odlyzko, A. On the Foundations of Combinatorial Theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42 (1973), 685–760.

[17] Rus, I. A., Iterates of Bernstein operators, via contraction principle. J. Math. Anal. Appl. 292 (2004), 259–261.

[18] Sablonniere, P. Positive Bernstein-Sheffer operators. J. Approx. Theory 83 (1995), 330–341.

[19] Stancu, D. D., Cismasiu, C. On an approximating linear positive operators of Cheney-Sharma. Rev. Anal. Numer. Theor. Approx. 26 (1997), nos. 1–2, 221–227.

[20] Stancu, D. D., Occorsio, M. R. On approximation by binomial operators of Tiberiu Popoviciu type. Rev. Anal. Numer. Theor. Approx. 27 (1998), no. 1, 167–181.

[21] Zeidler, E., Nonlinear Functional Analysis and Its Applications: Fixed-point Theorems. Transl. from the German by Peter R. Wadsack, Springer New York, 1993.


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