doi: 10.3934/mfc.2021042
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Generalized Kantorovich modifications of positive linear operators

1. 

Lucian Blaga University of Sibiu, Department of Mathematics and Informatics, Str. Dr. I. Ratiu, No.5-7, RO-550012 Sibiu, Romania

2. 

Technical University of Cluj-Napoca, Faculty of Automation and Computer Science, Department of Mathematics, Str. Memorandumului nr. 28, 400114 Cluj-Napoca, Romania

*Corresponding author: Ana-Maria Acu

Received  June 2021 Early access January 2022

Fund Project: This work was supported by a Hasso Plattner Excellence Research Grant (LBUS-HPI-ERG-2020-07), financed by the Knowledge Transfer Center of the Lucian Blaga University of Sibiu

Starting with a positive linear operator we apply the Kantorovich modification and a related modification. The resulting operators are investigated. We are interested in the eigenstructure, Voronovskaya formula, the induced generalized convexity, invariant measures and iterates. Some known results from the literature are extended.

Citation: Ana-Maria Acu, Ioan Cristian Buscu, Ioan Rasa. Generalized Kantorovich modifications of positive linear operators. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021042
References:
[1]

A. M. AcuA. Aral and I. Raşa, Generalized Bernstein Kantorovich operators: Voronovskaya type results, convergence in variation, Carpathian J. Math., 38 (2022), 1-12.  doi: 10.37193/cjm.2022.01.01.  Google Scholar

[2]

A. M. AcuM. Heilmann and I. Raşa, Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators Ⅱ, Positivity, 25 (2021), 1585-1599.  doi: 10.1007/s11117-021-00832-7.  Google Scholar

[3]

A. M. AcuM. Heilmann and I. Raşa, Iterates of convolution-type operators, Positivity, 25 (2021), 495-506.  doi: 10.1007/s11117-020-00773-7.  Google Scholar

[4]

A. M. AcuA.-I. Măduţa and I. Rasa, Voronovskaya type results and operators fixing two functions, Math. Model. Anal., 26 (2021), 395-410.  doi: 10.3846/mma.2021.13228.  Google Scholar

[5]

A. M. Acu and I. Raşa, Iterates and invariant measures for Markov operators, Results Math., 76 (2021), 218, 16pp. doi: 10.1007/s00025-021-01524-0.  Google Scholar

[6]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, Series: De Gruyter Studies in Mathematics, 17, 1994. doi: 10.1515/9783110884586.  Google Scholar

[7]

F. Altomare and I. Raşa, Lipschitz contractions, unique ergodicity and asymptotics of Markov semigroups, Boll. Unione Mat. Ital., 5 (2012), 1-17.   Google Scholar

[8]

A. Aral and H. Erbay, A note on the difference of positive operators and numerical aspects, Mediterr. J. Math., 17 (2020), Paper No. 45, 20 pp. doi: 10.1007/s00009-020-1489-5.  Google Scholar

[9]

A. AralD. Otrocol and ">I. Ras, On approximation by some Bernstein Kantorovich exponential-type polynomials, Period. Math. Hung., 79 (2019), 236-254.  doi: 10.1007/s10998-019-00284-3.  Google Scholar

[10]

D. Cárdenas-MoralesP. Garrancho and I. Rasa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 62 (2011), 158-163.  doi: 10.1016/j.camwa.2011.04.063.  Google Scholar

[11]

S. Cooper and S. Waldron, The eigenstructure of the Bernstein operator, J. Approx. Theory, 105 (2000), 133-165.  doi: 10.1006/jath.2000.3464.  Google Scholar

[12]

M. Heilmann and I. Rasa, Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators, Positivity, 21 (2017), 897-910.  doi: 10.1007/s11117-016-0441-1.  Google Scholar

[13]

R. P. Kelisky and T. J. Rivlin, Iterates of Bernstein polynomials, Pac. J. Math., 21 (1967), 511-520.  doi: 10.2140/pjm.1967.21.511.  Google Scholar

[14]

R. Păltănea, A note on generalized Benstein-Kantorovich operators, Bull. Transilv. Univ. Braşov Ser. Ⅲ, 6 (2013), 27-32.   Google Scholar

show all references

References:
[1]

A. M. AcuA. Aral and I. Raşa, Generalized Bernstein Kantorovich operators: Voronovskaya type results, convergence in variation, Carpathian J. Math., 38 (2022), 1-12.  doi: 10.37193/cjm.2022.01.01.  Google Scholar

[2]

A. M. AcuM. Heilmann and I. Raşa, Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators Ⅱ, Positivity, 25 (2021), 1585-1599.  doi: 10.1007/s11117-021-00832-7.  Google Scholar

[3]

A. M. AcuM. Heilmann and I. Raşa, Iterates of convolution-type operators, Positivity, 25 (2021), 495-506.  doi: 10.1007/s11117-020-00773-7.  Google Scholar

[4]

A. M. AcuA.-I. Măduţa and I. Rasa, Voronovskaya type results and operators fixing two functions, Math. Model. Anal., 26 (2021), 395-410.  doi: 10.3846/mma.2021.13228.  Google Scholar

[5]

A. M. Acu and I. Raşa, Iterates and invariant measures for Markov operators, Results Math., 76 (2021), 218, 16pp. doi: 10.1007/s00025-021-01524-0.  Google Scholar

[6]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, Series: De Gruyter Studies in Mathematics, 17, 1994. doi: 10.1515/9783110884586.  Google Scholar

[7]

F. Altomare and I. Raşa, Lipschitz contractions, unique ergodicity and asymptotics of Markov semigroups, Boll. Unione Mat. Ital., 5 (2012), 1-17.   Google Scholar

[8]

A. Aral and H. Erbay, A note on the difference of positive operators and numerical aspects, Mediterr. J. Math., 17 (2020), Paper No. 45, 20 pp. doi: 10.1007/s00009-020-1489-5.  Google Scholar

[9]

A. AralD. Otrocol and ">I. Ras, On approximation by some Bernstein Kantorovich exponential-type polynomials, Period. Math. Hung., 79 (2019), 236-254.  doi: 10.1007/s10998-019-00284-3.  Google Scholar

[10]

D. Cárdenas-MoralesP. Garrancho and I. Rasa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 62 (2011), 158-163.  doi: 10.1016/j.camwa.2011.04.063.  Google Scholar

[11]

S. Cooper and S. Waldron, The eigenstructure of the Bernstein operator, J. Approx. Theory, 105 (2000), 133-165.  doi: 10.1006/jath.2000.3464.  Google Scholar

[12]

M. Heilmann and I. Rasa, Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators, Positivity, 21 (2017), 897-910.  doi: 10.1007/s11117-016-0441-1.  Google Scholar

[13]

R. P. Kelisky and T. J. Rivlin, Iterates of Bernstein polynomials, Pac. J. Math., 21 (1967), 511-520.  doi: 10.2140/pjm.1967.21.511.  Google Scholar

[14]

R. Păltănea, A note on generalized Benstein-Kantorovich operators, Bull. Transilv. Univ. Braşov Ser. Ⅲ, 6 (2013), 27-32.   Google Scholar

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