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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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

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