Asymptotic properties of Kantorovich-type Szász–Mirakjan operators of higher order

Abstract


In this paper we study local approximation properties of a higher order Kantorovich-type Szász–Mirakjan operator recently introduced by Sabancigil, Kara, and Mahmudov. We derive the complete asymptotic expansion for these operators. They generalize the Szász–Mirakjan operators of Kantorovich-type and approximate locally integrable functions satisfying a certain growth condition on the infinite interval \(0,\infty\).

Authors

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

Octavian Agratini
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Street Fântânele, nr. 57, 400320 Cluj-Napoca, Romania

Mircea Ivan
Technical University of Cluj-Napoca, Str. Memorandumului nr. 28, 400114 Cluj-Napoca, Romania

Keywords

Linear positive operator; Szasz operator; Kantorovich operator; Stirling numbers.

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U. Abel, O. Agratini, M. Ivan, Asymptotic properties of Kantorovich-type Szász–Mirakjan operators of higher order, Mathematical Foundations of  Computing, 2023, https://doi.org/10.3934/mfc.2023003

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Journal

Mathematical Foundations of Computing

Publisher Name

Mathematical Foundations of Computing

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Online ISSN

2577-8838

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References

[1] U. Abel, Asymptotic approximation with Kantorovich polynomial, Approx. Theory Appl. (N.S.), 14 (1998), 106-116. https://doi.org/10.1007/BF02836771
[2] U. Abel and M. Ivan, Asymptotic expansion of the Jakimovski-Leviatan operators and their derivatives, in Functions, Series, Operators (Budapest, 1999), János Bolyai Math. Soc., Budapest, 2002,103-119.
[3] A.-M. Acu, I. C. Buscu and I. Rasa, Generalized Kantorovich modifications of positive linear operators, Mathematical Foundations of Computing, 6 (2023), 54-62,  https://doi.org/10.3934/mfc.2021042
[4] O. Agratini, Uniform approximation of some classes of linear positive operators expressed by series, Appl. Anal., 94 (2015), 1662-1669.  https://doi.org/10.1080/00036811.2014.940919.
[5] F. Altomare, Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory, 5 (2010), 92-164.
[6] F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, vol. 17 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1994, Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff. https://doi.org/10.1515/9783110884586.
[7] M. Becker, Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27 (1978), 127-142, https://doi.org/10.1512/iumj.1978.27.27068.
[8] M. Becker, D. Kucharski and R. J. Nessel, Global approximation theorems for the Szász-Mirakjan operators in exponential weight spaces, in Linear Spaces and Approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1977), Internat. Ser. Numer. Math., Vol. 40, Birkhäuser, Basel, 1978,319-333.
[9] H. Bohman, On approximation of continuous and of analytic functions, Ark. Mat., 2 (1952), 43-56, https://doi.org/10.1007/BF02591381.
[10] B. D. Boyanov and V. M. Veselinov, A note on the approximation of functions in an infinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 14 (1970), 9-13 (1971).
[11] J. Bustamante, J. M. Quesada and L. Morales de la Cruz, Direct estimate for positive linear operators in polynomial weighted spaces, J. Approx. Theory, 162 (2010), 1495-1508, https://doi.org/10.1016/j.jat.2010.04.001.
[12] P. L. Butzer, On the extensions of Bernstein polynomials to the infinite interval, Proc. Amer. Math. Soc., 5 (1954), 547-553, https://doi.org/10.1090/S0002-9939-1954-0063483-7.
[13] L. Comtet, Advanced Combinatorics, enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974, The Art of Finite and Infinite Expansions.
[14] J. de la Cal and J. Cárcamo, On uniform approximation by some classical Bernstein-type operators, J. Math. Anal. Appl., 279 (2003), 625-638, https://doi.org/10.1016/S0022-247X(03)00048-9.
[15] Z. Ditzian, Convergence of sequences of linear positive operators: Remarks and applications, J. Approximation Theory, 14 (1975), 296-301, https://doi.org/10.1016/0021-9045(75)90076-3.
[16] Z. Ditzian and V. Totik, Moduli of Smoothness, vol. 9 of Springer Series in Computational Mathematics, Springer-Verlag, New York, 1987, https://doi.org/10.1007/978-1-4612-4778-4.
[17] J. Favard, Sur les multiplicateurs d’interpolation, J. Math. Pures Appl. (9), 23 (1944), 219-247.
[18] H. Feng, S. Hou, L.-Y. Wei and D.-X. Zhou, CNN models for readability of Chinese texts, Mathematical Foundations of Computing, 5 (2022), 351-362, https://doi.org/10.3934/mfc.2022021.
[19] A. D. Gadžiev, Theorems of the type of P. P. Korovkin’s theorems, Mat. Zametki, 20 (1976), 781-786.
[20] V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, Cham, 2014, https://doi.org/10.1007/978-3-319-02765-4.

2023

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