## Banach Journal of Mathematical Analysis

### A generalization of Kantorovich operators for convex compact subsets

#### Abstract

In this article, we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number, and a sequence of Borel probability measures. By considering special cases of these parameters for particular convex compact subsets, we obtain the classical Kantorovich operators defined in the $1$-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, we discuss the preservation of Lipschitz-continuity and of convexity.

#### Article information

Source
Banach J. Math. Anal., Volume 11, Number 3 (2017), 591-614.

Dates
Accepted: 19 September 2016
First available in Project Euclid: 6 May 2017

https://projecteuclid.org/euclid.bjma/1494036023

Digital Object Identifier
doi:10.1215/17358787-2017-0008

Mathematical Reviews number (MathSciNet)
MR3679897

Zentralblatt MATH identifier
1383.41003

Subjects
Primary: 47B65: Positive operators and order-bounded operators
Secondary: 41A36: Approximation by positive operators

#### Citation

Altomare, Francesco; Cappelletti Montano, Mirella; Leonessa, Vita; Raşa, Ioan. A generalization of Kantorovich operators for convex compact subsets. Banach J. Math. Anal. 11 (2017), no. 3, 591--614. doi:10.1215/17358787-2017-0008. https://projecteuclid.org/euclid.bjma/1494036023

#### References

• [1] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975.
• [2] F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, de Gruyter Stud. Math. 17, de Gruyter, Berlin, 1994.
• [3] F. Altomare, M. Cappelletti Montano, and V. Leonessa, On a generalization of Kantorovich operators on simplices and hypercubes, Adv. Pure Appl. Math. 1 (2010), no. 3, 359–385.
• [4] F. Altomare, M. Cappelletti Montano, V. Leonessa, and I. Raşa, Markov Operators, Positive Semigroups and Approximation Processes, de Gruyter Stud. Math. 61, de Gruyter, Berlin, 2014.
• [5] F. Altomare, M. Cappelletti Montano, V. Leonessa, and I. Raşa, On differential operators associated with Markov operators, J. Funct. Anal. 266 (2014), no. 6, 3612–3631.
• [6] F. Altomare and V. Leonessa, On a sequence of positive linear operators associated with a continuous selection of Borel measures, Mediterr. J. Math. 3 (2006), no. 3–4, 363–382.
• [7] H. Berens and R. DeVore, Quantitative Korovkin theorems for positive linear operators on $L_{p}$-spaces, Trans. Amer. Math. Soc. 245 (1978), 349–361.
• [8] J. de la Cal and A. M. Valle, A generalization of Bernstein-Kantorovič operators, J. Math. Anal. Appl. 252 (2000), no. 2, 750–766.
• [9] J. Edwards, A Treatise on the Integral Calculus with Applications, Examples and Problems, Vols. 1, 2, Macmillan, London, 1921. JFM48.1196.02.
• [10] H. Gonska, I. Raşa, and M.-D. Rusu, Applications of an Ostrowski-type inequality, J. Comput. Anal. Appl. 14 (2012), no. 1, 19–31.
• [11] L.V. Kantorovich, Sur certains développements suivant les polynômes de la forme de B. Bernstein, I, II, C.R. Acad. URSS 1930, 563–568, 595–600.
• [12] J. Nagel, Kantorovič operators of second order, Monatsh. Math. 95 (1983), no. 1, 33–44.
• [13] V. A. Popov, One-sided approximation of periodic functions of several variables, C. R. Acad. Bulgare Sci. 35 (1982), no. 12, 1639–1642.
• [14] E. Quak, Multivariate $L_{p}$-error estimates for positive linear operators via the first-order $\tau$-modulus, J. Approx. Theory 56 (1989), no. 3, 277–286.
• [15] T. Sauer, Multivariate Bernstein polynomials and convexity, Comput. Aided Geom. Design 8 (1991), no. 6, 465–478.
• [16] B. Sendov and V. A. Popov, The Averaged Moduli of Smoothness: Applications in Numerical Methods and Approximation, Pure Appl. Math. (N. Y.), Wiley, Chichester, 1988.
• [17] D. D. Stancu, On a generalization of Bernstein polynomials, Stud. Univ. Babeş-Bolyai Math. 14 (1969), no. 2, 31–45.
• [18] D. X. Zhou, Converse theorems for multidimensional Kantorovich operators, Anal. Math. 19 (1993), no. 1, 85–100.