On the variation detracting property of operators of Balázs and Szabados


In this note we spotlight the linear and positive operators of discrete type \((R_{n})_{n\geq1}\) known as Balazs–Szabados operators. We prove that this sequence enjoys the variation detracting property. The convergence in variation of \((R_{n}f)_{n\geq1}\) to f is also proved. A generalization in Kantorovich sense is constructed and boundedness with respect to BV-norm is revealed.


Ulrich Abel
Fachbereich MND, Technische Hochschule Mittelhessen, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania


linear positive operator; function of bounded variation; variation detracting property

Paper coordinates

U. Abel, O. Agratini, On the variation detracting property of operators of Balázs and Szabados, Acta Mathematica Hungarica, 150 (2016), pp. 383-395, https://doi.org/10.1007/s10474-016-0642-x


About this paper


Acta Mathematica Hungarica

Publisher Name


Print ISSN
Online ISSN

google scholar link

[1] U. Abel and B. Della Vecchia, Asymptotic approximation by the operators of K. Balazs and Szabados, Acta Sci. Math. (Szeged), 66 (2000), 137–145.
[2] J. A. Adell and J. de la Cal, Bernstein-type operators diminish the φ-variation, Constructive Approx., 12 (1996), 489–507.
[3] O. Agratini, On approximation properties of Balazs–Szabados operators and their Kantorovich extension, Korean J. Comput. & Appl. Math., 9 (2002), 361–372.
[4] O. Agratini, On the variation detracting property of a class of operators, Applied Mathematics Letters, 19 (2006), 1261–1264.
[5] K. Balazs, Approximation by Bernstein rational functions, Acta Math. Acad. Sci. Hungar., 26 (1975), 123–134.
[6] C. Balazs and J. Szabados, Approximation by Bernstein type rational functions. II, Acta Math. Acad. Sci. Hungar., 40 (1982), 331–337. Acta Mathematica Hungarica
[7] C. Bardaro, P. L. Butzer, R. L. Stens and G. Vinti, Convergence in variation and rate of approximation for Bernstein-type polynomials and singular convolution integrals, Analysis, 23 (2003), 299–340.
[8] B. Della Vecchia, On some preservation and asymptotic relations of a rational operator, Facta Universitatis (Nis), Mathematics and Informatics, 4 (1989), 57–62.
[9] H. Karsli, On convergence of Chlodovsky and Chlodovsky–Kantorovich polynomials in the variation seminorm, Mediterranean J. Math., 10 (2013), 41–56.
[10] A. Kivinukk and T. Metsm¨agi, Approximation in variation by the Meyer–Konig and Zeller operators, Proceedings of the Estonian Academy of Sciences, 60 (2011), 88–97.
[11] G. G. Lorentz, Bernstein Polynomials, University of Toronto Press (Toronto, 1953); 2nd Edition, Chelsea Publishing Co. (New York, 1986).
[12] S. Micula, On spline collocation and the Hilbert transform, Carpathian J. Math., 31 (2015), 89–95.
[13] O. Oksuzer, H. Karsli and F. Ta¸sdelen Ye¸sildal, On convergence of Bernstein–Stancu polynomials in the variation seminorm, Numerical Functional Analysis and Optimization, online: 29 Jan. 2016, DOI: 10.1080/01630563.2015.1137938, pp. 23.
[14] J. Peetre, A theory of interpolation of normed spaces, Notas de Matematica, Instituto de Matematica Pura e Aplicada, Rio de Janeiro, 39 (1968), 1–86.
[15] V. Totik, Saturation for Bernstein type rational functions, Acta Math. Hungar., 43 (1984), 219–250.


Related Posts