Modified positive linear operators, iterates and systems of linear equations

Ioan Cristian Buşcu; Gabriela Motronea; Vlad Paşca

doi:10.3934/mfc.2023031

Article Contents

2024, Volume 7, Issue 4: 568-577. Doi: 10.3934/mfc.2023031

This issue Previous Article Approximation properties of Riemann-Liouville type fractional Bernstein-Kantorovich operators of order $ \alpha $ Next Article Smoothing piecewise linear activation functions based on mollified square root functions

Modified positive linear operators, iterates and systems of linear equations

1.
Technical University of Cluj-Napoca, Romania
2.
Lucian Blaga University of Sibiu, Romania

^*Corresponding author: Vlad Paşca
^*Corresponding author: Vlad Paşca

Received: March 2023

Revised: June 2023

Early access: July 10, 2023

Published online: July 10, 2023

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Abstract

We consider Kantorovich modifications of linking operators and Stancu modifications of the classical Bernstein operators. For the modified operators we determine the limits of iterates and the invariant measures. In order to find the limits we have to solve systems of linear equations and to this end we use a suitable iterative algorithm.

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Mathematics Subject Classification: Primary: 41A36; Secondary: 65H10.

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References

[1]	A. Acu, I. Buscu and I. Rasa, Generalized Kantorovich modifications of positive linear operators, Mathematical Foundations of Computing, 6 (2023), 54-62. doi: 10.3934/mfc.2021042.
[2]	A. M. Acu, M. Heilmann, I. Rasa and A. Seserman, Poisson approximation to the binomial distribution: Extensions to the convergence of positive operators, arXiv: 2208.08326.
[3]	A. M. Acu, H. Heilmann and I. Rasa, Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators Ⅱ, Positivity, 25 (2021), 1585-1599. doi: 10.1007/s11117-021-00832-7.
[4]	A. M. Acu and I. Rasa, Nonlinear algebraic systems with positive coefficients and positive solutions, J. Appl. Math. Comput., 69 (2023), 19-35. doi: 10.1007/s12190-022-01732-z.
[5]	A. M. Acu, I. Rasa and A. Seserman, Positive linear operators and exponential functions, Mathematical Foundations of Computing, 6 (2022), 1-7. doi: 10.3934/mfc.2022050.
[6]	A. M. Acu, I. Raşa and A. E. Şteopoaie, Algebraic systems with positive coefficients and positive solutions, Mathematics, 10 (2022), 1327.
[7]	O. Agratini, Aproximare Prin Operatori Liniari, Presa Universitară Clujeană, Cluj-Napoca, 2000.
[8]	F. Altomare, M. Campiti, Korovkin-type Approximation Theory and its Applications, De Gruyter Stud. Math., 17. Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.
[9]	A. Ciurte, S. Nedevschi and I. Rasa, A generalization of the EMML and ISRA algorithms for solving linear systems, J. Comput. Anal. Appl., 12 (2010), 799-816.
[10]	A. Ciurte, S. Nedevschi and I. Rasa, Systems of nonlinear algebraic equations with unique solution, Numer. Algor., 68 (2015), 367-376. doi: 10.1007/s11075-014-9849-5.
[11]	L. Coroianu, D. Costarelli, S. G. Gal and G. Vinti, Approximation by max-product sampling Kantorovich operators with generalized kernels, Anal. Appl., 19 (2021), 219-244. doi: 10.1142/S0219530519500155.
[12]	J. de la Cal and F. Luquin, Approximation Szász and Gamma operators by Baskakov operators, J. Math. Anal. Appl., 184 (1994), 585-593. doi: 10.1006/jmaa.1994.1223.
[13]	H. H. Gonska and J. Meier, Quantitative theorems on approximation by Bernstein-Stancu operators, Calcolo, 21 (1984), 317-335. doi: 10.1007/BF02576170.
[14]	H. Gonska, P. Piţul and I. Raşa, Over-iterates of Bernstein-Stancu operators, Calcolo, 44 (2007), 117-125. doi: 10.1007/s10092-007-0131-2.
[15]	I. Györi, F. Hartung and N. A.Mohamady, Existence and uniqueness of positive solutions of a system of nonlinear algebraic equations, Period. Math. Hung., 75 (2017), 114-127. doi: 10.1007/s10998-016-0179-3.
[16]	M. Heilmann and I. Raşa, A Nice Representation for a Link between Bernstein-Durrmeyer and Kantorovich Operators, Commun. Comput. Inf. Sci., 655. Springer, Singapore, 2017.
[17]	M. Kaykobad, Positive solutions of positive linear systems, Linear Algebra Appl., 64 (1985), 133-140. doi: 10.1016/0024-3795(85)90271-X.
[18]	A. Kumar, A new variant of the modified Bernstein-Kantorovich operators defined by Özarslan and Duman, Mathematical Foundations of Computing, 6 (2023), 1-23. doi: 10.3934/mfc.2022062.
[19]	G. J. McLachlan and T. Krishnan, The EM Algorithm and Extensions, Wiley Ser. Probab. Stat. Wiley-Interscience, Hoboken, NJ, 2008. doi: 10.1002/9780470191613.
[20]	V. Paşca, A. Seserman and A. E. Şteopoaie, Iterates of positive linear operators and linear systems of equations, submitted.
[21]	R. Schnabl, Zum globalen Saturationsproblem der Folge der Bernstein-operatoren, Acta Sci. Math., 31 (1970), 351-358.
[22]	D. D. Stancu, Approximation of functions by means of some new classes of positive linear operators, Numerische Methoden der Approximations-theorie Bd., Band 1, Internat. Schriftenreihe Numer. Math., Band 16, Birkhäuser, Basel, (1972), 187-203.
[23]	S.M. Stefanov, Numerical solution of some systems of nonlinear algebraic equations, J. Interdiscip. Math., 24 (2021), 1545-1564. doi: 10.1080/09720502.2020.1833462.