The aim of this article is to present a general class of positive linear operators of discrete type related to squared fundamental basis functions. If these operators are expressed by a series, we propose to truncate them by a finite sum while still keeping the property of converging towards the identity operator in a weighted space. A relation between the local smoothness of functions and the local approximation is obtained. In the variant in which the operators are described by a finite sum, we establish the limit of their iterates.


Octavian Agratini
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania


Contraction principle; linear positive operator; Lipschitz function; modulus of smoothness; weighted space

Paper coordinates

O. Agratini, Properties of positive linear operators connected with squared fundamental functions, Numer. Funct. Anal. Optimiz., 45 (2024) no. 2, pp. 103–111https://doi.org/10.1080/01630563.2024.2316579


About this paper


Numerical Functional Analysis and Optimization

Publisher Name

Taylor and Francis Ltd.

Print ISSN


Online ISSN


google scholar link

[1] Herzog, F. (2009). Heuristische untersuchungen der konwergenzrate spazieller linearerapproximationsprozesse. Diploma Thesis, Friedberg.
[2] Gavrea, I., Ivan, M. (2017). On a new sequence of positive linear operators related tosquared Bernstein polynomials.Positivity21:911–917.
[3] Holhos, A. (2019). Voronovskaya theorem for a sequence of positive linear operatorsrelatedtosquaredBernsteinpolynomials.Positivity23:571–580.
[4] Abel, U., Kushnirevych, V. (2019). Voronovskaja type theorems for positive linear opera-tors related tosquare d Bernstein polynomials. Positivity23:697–710.
[5] Harremoes, P., Topsoe, F. (2001). Inequalities between entropy and index of coincidence derived from in formation diagrams.IEEE Trans. Inf. Theory47(7):2944–2960.
[6] Abel, U. (2020). Voronovskaja type theorems for positive linear operators related to squared fundamental functions.In:Draganov,B.,Ivanov,K.,Nikolov, G.,Uluchev,
R.,eds. Constructive Theory of Functions, Sozopol 2019. Sofia: Prof. Marin Drinov PublishingHouse of BAS, pp. 1–21.
[7] Agratini, O., Tarabie, S., Trîmbitas, R. (2014). Approximation of bivariate function bytruncated classes of operators, In: Simian, D., ed.Proceedings of the Third International Conference on Modelling and Development of Intelligent Systems, Sibiu 2013. Sibiu:Lucian Blaga University Press, pp. 11–19.
[8] Lehnhoff, H.-G. (1984). On a modified Szász-Mirakjan operator.J. Approx. Theory42:278–282.
[9] Wang, J., Zhou, S. (2000). On the convergence of modified Baskakov operators. Bull.  Inst. Math. Academia Sinica 28(2):117–123.
[10] Shisha, O., Mond, B. (1968). The degree of convergence of linear positive operators.Proc.Nat. Acad. Sci. USA60:1196–1200.
[11] Agratini, O., Rus, I. A. (2003). Iterates of a class of discrete operators via contraction principle. Comment. Math. Univ. Carolinae 44(3):555–563.


Related Posts