79636158-3032-42ff-866d-54bf3ae1684d20250212081710219wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40511220241220242310.37394/23206.2024.23https://wseas.com/journals/mathematics/2024.phpOctavianAgratiniTiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Street Fântânele 57, 400320 Cluj-Napoca, ROMANIAThis note is devoted to the study of a linear positive sequence of operators representing an integral form in Kantorovich's sense. We prove that this sequence converges to the identity operator in Lp([0,1]), p ≥ 1, spaces. By using the r-th order (r = 1 and r ≥ 3) modulus of smoothness measured in these spaces, we establish an upper bound of the approximation error. Also, we point out a connection between the smoothness of α- Hölder (0 < α ≤ 1) functions and the local approximation property.123120241231202410331038106https://wseas.com/journals/mathematics/2024/c145106-054(2024).pdf10.37394/23206.2024.23.106https://wseas.com/journals/mathematics/2024/c145106-054(2024).pdfL.V. Kantorovich, Sur certains développements suivant les polynômes de la forme de S. Bernstein, I, II, C.R. Acad. URSS, (1930), 563-568, 595-600. 10.1090/s0002-9939-1954-0063483-7P. L. Butzer, On the extensions of Bernstein polynomials to the infinite interval, Proc. Amer. Math. Soc., 5(1954), 547-553. Z. Ditzian, V. Totik, Moduli of Smoothness, Springer-Verlag, New York Inc., 1987. Q. Razi, Approximation of a function by Kantorovich type operators, Math. Vesnik, vol. 41(1989), No. 3, 183-192, [Online]. http://eudml.org/doc/260465 (Accessed Date: October 5, 2024). A. Habib, A. Wafi, Degree of approximation of functions by modified Bernstein polynomials on an unbounded interval, Indian J. Pure Appl. Math., 8(6)(1977), 691-695. F. Altomare, M. Cappelletti Montano, V. Leonessa, Kantorovich-type modifications of certain discrete-type operators on the positive real axis, Note Mat., 43(2023), no. 1, 15-40. DOI: 10.1285/i15900932v43n1p15. 10.1515/math-2022-0573L. Angeloni, G. Vinti, Multidimensional sampling-Kantorovich operators in BVspaces, Open Math., 21(2023), no. 1, pp. 20220573. http://dx.doi.org/10.1515/math2022-0573. M. Kara, N. I. Mahmudov, Approximation theorems for complex -BernsteinKantorovich operators, Results Math., 79(2024), art. no. 72. https://doi.org/10.1007/s00025-023-02101-3. 10.18514/mmn.2001.31O. Agratini, An approximation process of Kantorovich type, Miskolc Math. Notes, 2(2001),. No. 1, 3-10. https://doi.org/10.18514/MMN.2001.31. 10.1007/s00009-006-0084-8F. Altomare, V. Leonessa, On a sequence of positive linear operators associated with a continuous selection of Borel measures, Mediterr. J. Math., 3(2006), 363-382. https://doi.org/10.1007/s00009-006-0084-8. Ö. Dalmanoglu, O. Dogru, On statistical approximation properties of Kantorovich type q-Bernstein operators, Math. Comput. Model., 52(2010), 760-771. https://doi.org/10.1016/j.mcm.2010.05.005. 10.1186/1029-242x-2014-10M.-Y. Ren, X.-M. Zeng, Some statistical approximation properties of Kantorovich-type q-Bernstein-Stancu operators, J. Inequal. Appl., 2014(2014), art. no. 10. https://doi.org/10.1186/1029-242X-2014-10. 10.1007/bf02591381H. Bohman, On approximation of continuous and of analytic functions, Ark. Math., 2(1952), No. 1, 43-56. P. P. Korovkin, Convergence of linear positive operators in the space of continuous functions (Russian), Dokl. Akad. Nauk SSSR, 90(1953), 961-964. 10.15672/hjms.20164515994O. Agratini, Kantorovich-type operators preserving affine functions, Hacet. J. Math. Stat., 45(2016), Issue 6, 1957-1663. DOI : 10.15672/HJMS.20164515994. 10.1007/978-3-662-02888-9_10R. A. DeVore, G. G. Lorentz, Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, 303, Springer-Verlag, Berlin, Heidelberg, 1993. 10.1515/9783110884586.141F. Altomare, Korovkin-type theorems and approximation by positive linear operators, Surveys in Approximation Theory, 5(2010), 92-164. J. Peetre, A Theory of Interpolation of Normed Spaces, Notes Universidade de Brasilia, 1963. H. Johnen, Inequalities connected with the moduli of smoothness, Mat. Vesnik, 24(1972), No. 1, 289-305. E. M. Stein, Singular Integrals and Differentia-bility Properties of Functions, Princeton University Press, 1970. J. J. Swetits, B. Wood, Note on the degree of Lp approximation with positive linear operators, J. Approx. Theory, 87(1996), 239- 241, art. no. 0103. https://doi.org/10.1006/jath.1996.0103. 10.1112/plms/s3-23.1.1S. Goldberg, A. Meir, Minimum moduli of ordinary differential operators, Proc. London Math. Soc., 23(1971), No. 1, 1-15.