A Sequence of Kantorovich-Type Operators on Mobile Intervals

Abstract

In this paper, we introduce and study a new sequence of positive linear operators, acting on both spaces of continuous functions as well as spaces of integrable functions on $[0, 1]$. We state some qualitative properties of this sequence and we prove that it is an approximation process both in $C([0, 1])$ and in $L^p([0, 1])$, also providing  some estimates of the rate of convergence. Moreover, we determine an asymptotic formula and, as an application,  we  prove that certain iterates of the operators converge, both in $C([0, 1])$ and, in some cases,  in $L^p([0, 1])$, to a limit semigroup. Finally, we show that our operators, under suitable hypotheses, perform better than  other existing ones in the literature. 

Keywords

• Kantorovich-type operators,Positive approximation processes,Rate of convergence,Asymptotic formula,Generalized convexity

References

1. [1] T. Acar, A. Aral, I. Ras ̧a, Positive linear operators preserving τ and τ2, Constr. Math. Anal. 2 (3) (2019), 98–102.
2. [2] T.Acar,M.CappellettiMontano,P.Garrancho,V.Leonessa,OnsequencesofJ.P.King-typeoperators,J.Funct.Spaces, 2019, Article ID 2329060.
3. [3] F. Altomare, M. Campiti, Korovkin-type approximation theory and its applications, de Gruyter Studies in Mathe- matics 17, Walter de Gruyter & Co., Berlin, 1994.
4. [4] F. Altomare, M. Cappelletti Montano, V. Leonessa, On a generalization of Kantorovich operators on simplices and hypercubes, Adv. Pure Appl. Math. 1(3) (2010), 359-385.
5. [5] F. Altomare, M. Cappelletti Montano, V. Leonessa, Iterates of multidimensional Kantorovich-type operator and their associated positive C0-semigroups, Studia Universitatis Babes-Bolyai. Mathematica 56(2) (2011), 236-251.
6. [6] F. Altomare, M. Cappelletti Montano, V. Leonessa, I. Ras ̧a, Markov Operators, Positive Semigroups and Approximation Processes, de Gruyter Studies in Mathematics 61, Walter de Gruyter GmbH, Berlin/Boston, 2014.
7. [7] F. Altomare, V. Leonessa, On a sequence of positive linear operators associated with a continuous selection of Borel mea- sures, Mediterr. J. Math. 3 (2006), 363-382.
8. [8] H. Bauer, Probability Theory, de Gruyter Studies in Mathematics 23, Walter de Gruyter & Co., Berlin, 1996.

9. [9] D. Cardenas-Morales, P. Garrancho, F.J. Muños-Delgado, Shape preserving approximation by Bernstein-type operators which fix polinomials, Appl. Math. Comput. 182 (2006) 1615–1622.
10. [10] D. Cardenas-Morales, P. Garrancho, I. Raşa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl. 62(1) (2011), 158–163.
11. [11] G. Freud, On approximation by positive linear methods I, II, Stud. Scin. Math. Hungar. 2 (1967) 63–66, 3 (1968), 365– 370.
12. [12] H. Gonska, P. Pitul, I. Raşa, General King-type operators, Results Math. 53(3-4) (2009) 279–286.
13. [13] J.P. King, Positive linear operators which preserve x2, Acta Math. Hungar. 99(3) (2003) 203–208.
14. [14] A.-J. López-Moreno, F.-J. Muñoz-Delgado, Asymptotic expression of derivatives of Bernstein type operators, Suppl. Cir. Mat. Palermo Ser. II 68 (2002), 615-624.
15. [15] R. Paltanea, Approximation theory using positive linear operators, Birkhäuser, Boston, (2004).
16. [16] J. J. Swetits, B. Wood, Quantitative estimates for Lp approximation with positive linear operators, J. Approx. Theory 38 (1983), 81–89.
17. [17] Z. Ziegler, Linear approximation and generalized convexity, J. Approx. Theory 1 (1968), 420–433.