August  2022, 5(3): 259-268. doi: 10.3934/mfc.2021034

New approximation properties of the Bernstein max-min operators and Bernstein max-product operators

Department of Mathematics and Computer Science, University of Oradea, Universitatii 1, 410087, Oradea, Romania

* Corresponding author: Lucian Coroianu

Received  July 2021 Revised  October 2021 Published  August 2022 Early access  November 2021

In this paper we put in evidence localization results for the so-called Bernstein max-min operators and a property of translation for the Bernstein max-product operators.

Citation: Lucian Coroianu, Sorin G. Gal. New approximation properties of the Bernstein max-min operators and Bernstein max-product operators. Mathematical Foundations of Computing, 2022, 5 (3) : 259-268. doi: 10.3934/mfc.2021034
References:
[1]

A. G. AnastassiouL. Coroianu and S. G. Gal, Approximation by a nonlinear Cardaliaguet-Euvrard neural network operator of max-product kind, J. Comp. Anal. Appl., 12 (2010), 396-406. 

[2]

B. BedeL. Coroianu and S. G. Gal, Approximation and shape preserving properties of the Bernstein operator of max-product kind, Intern. J. Math. Math. Sci., 2009 (2009), 590589.  doi: 10.1155/2009/590589.

[3]

B. BedeL. Coroianu and S. G. Gal, Approximation and shape preserving properties of the nonlinear Meyer-König and Zeller operator of max-product kind, Numer. Funct. Anal. Optim., 31 (2010), 232-253.  doi: 10.1080/01630561003757686.

[4]

B. Bede, L. Coroianu and S. G. Gal, Approximation by Max-Product Type Operators, Springer, [Cham], 2016. doi: 10.1007/978-3-319-34189-7.

[5]

B. BedeH. NobuharaJ. Fodor and K. Hirota, Max-product Shepard approximation operators,, J. Adv. Comput. Intell. Inform., 10 (2006), 494-497.  doi: 10.20965/jaciii.2006.p0494.

[6]

L. CoroianuD. CostarelliS. G. Gal and G. Vinti, Approximation by max-product sampling Kantorovich operators with generalized kernels, Anal. Appl. (Singap.), 19 (2021), 219-244.  doi: 10.1142/S0219530519500155.

[7]

L. CoroianuD. CostarelliS. G. Gal and G. Vinti, Connections between the approximation orders of positive linear operators and their max-product counterparts, Numer. Funct. Anal. Optim., 42 (2021), 1263-1286.  doi: 10.1080/01630563.2021.1954018.

[8]

L. CoroianuD. CostarelliS. G. Gal and G. Vinti, The max-product generalized sampling operators: Convergence and quantitative estimates,, Appl. Math. Comput., 355 (2019), 173-183.  doi: 10.1016/j.amc.2019.02.076.

[9]

L. Coroianu and S. G. Gal, Approximation by max-product Lagrange interpolation operators, Stud. Univ. Babeş -Bolyai Math., 56 (2011), 315-325. 

[10]

L. Coroianu and S. G. Gal, Classes of functions with improved estimates in approximation by the max-product Bernstein operator,, Anal. Appl. (Singap.), 9 (2011), 249-274.  doi: 10.1142/S0219530511001856.

[11]

L. Coroianu and S. G. Gal, Localization results for the Bernstein max-product operator,, Appl. Math. Comput., 231 (2014), 73-78.  doi: 10.1016/j.amc.2013.12.190.

[12]

L. CoroianuS. G. Gal and B. Bede, Approximation of fuzzy numbers by Bernstein operators of max-product kind,, Fuzzy Set. Syst., 257 (2014), 41-66.  doi: 10.1016/j.fss.2013.04.010.

[13]

D. CostarelliA. R. Sambucini and G. Vinti, Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications,, Neural Comput. Appl., 31 (2019), 5069-5078.  doi: 10.1007/s00521-018-03998-6.

[14]

D. Costarelli and G. Vinti, Max-product neural network and quasi-interpolation operators activated by sigmoidal functions,, J. Approx. Theory, 209 (2016), 1-22.  doi: 10.1016/j.jat.2016.05.001.

[15]

S. G. Gal, Shape-Preserving Approximation by Real and Complex Polynomials, Birkhäuser, Boston-Basel-Berlin, 2008. doi: 10.1007/978-0-8176-4703-2.

[16]

T. Y. Gökçer and O. Duman, Approximation by max-min operators: A general theory and its applications,, Fuzzy Sets and Systems, 394 (2020), 146-161.  doi: 10.1016/j.fss.2019.11.007.

[17]

T. Y. Gökcer and O. Duman, Summation process by max-product operators,, Computational Analysis, 155 (2016), 59-67.  doi: 10.1007/978-3-319-28443-9_4.

[18]

S. Y. Güngör and N. Ispir, Approximation by Bernstein-Chlodowsky operators of max-product kind., Math. Commun., 23 (2018), 205-225. 

[19]

A. Holhoş, Weighted approximation of functions by Favard operators of max-product type,, Period. Math. Hungar., 77 (2018), 340-346.  doi: 10.1007/s10998-018-0249-9.

[20]

A. Holhoş, Weighted approximation of functions by Meyer-K önig and Zeller operators of max-product type,, Numer. Funct. Anal. Optim., 39 (2018), 689-703.  doi: 10.1080/01630563.2017.1413386.

[21]

S. Karakus and K. Demirci, Statistical $\sigma $-approximation to max-product operators,, Comput. Math. Appl., 61 (2011), 1024-1031.  doi: 10.1016/j.camwa.2010.12.052.

show all references

References:
[1]

A. G. AnastassiouL. Coroianu and S. G. Gal, Approximation by a nonlinear Cardaliaguet-Euvrard neural network operator of max-product kind, J. Comp. Anal. Appl., 12 (2010), 396-406. 

[2]

B. BedeL. Coroianu and S. G. Gal, Approximation and shape preserving properties of the Bernstein operator of max-product kind, Intern. J. Math. Math. Sci., 2009 (2009), 590589.  doi: 10.1155/2009/590589.

[3]

B. BedeL. Coroianu and S. G. Gal, Approximation and shape preserving properties of the nonlinear Meyer-König and Zeller operator of max-product kind, Numer. Funct. Anal. Optim., 31 (2010), 232-253.  doi: 10.1080/01630561003757686.

[4]

B. Bede, L. Coroianu and S. G. Gal, Approximation by Max-Product Type Operators, Springer, [Cham], 2016. doi: 10.1007/978-3-319-34189-7.

[5]

B. BedeH. NobuharaJ. Fodor and K. Hirota, Max-product Shepard approximation operators,, J. Adv. Comput. Intell. Inform., 10 (2006), 494-497.  doi: 10.20965/jaciii.2006.p0494.

[6]

L. CoroianuD. CostarelliS. G. Gal and G. Vinti, Approximation by max-product sampling Kantorovich operators with generalized kernels, Anal. Appl. (Singap.), 19 (2021), 219-244.  doi: 10.1142/S0219530519500155.

[7]

L. CoroianuD. CostarelliS. G. Gal and G. Vinti, Connections between the approximation orders of positive linear operators and their max-product counterparts, Numer. Funct. Anal. Optim., 42 (2021), 1263-1286.  doi: 10.1080/01630563.2021.1954018.

[8]

L. CoroianuD. CostarelliS. G. Gal and G. Vinti, The max-product generalized sampling operators: Convergence and quantitative estimates,, Appl. Math. Comput., 355 (2019), 173-183.  doi: 10.1016/j.amc.2019.02.076.

[9]

L. Coroianu and S. G. Gal, Approximation by max-product Lagrange interpolation operators, Stud. Univ. Babeş -Bolyai Math., 56 (2011), 315-325. 

[10]

L. Coroianu and S. G. Gal, Classes of functions with improved estimates in approximation by the max-product Bernstein operator,, Anal. Appl. (Singap.), 9 (2011), 249-274.  doi: 10.1142/S0219530511001856.

[11]

L. Coroianu and S. G. Gal, Localization results for the Bernstein max-product operator,, Appl. Math. Comput., 231 (2014), 73-78.  doi: 10.1016/j.amc.2013.12.190.

[12]

L. CoroianuS. G. Gal and B. Bede, Approximation of fuzzy numbers by Bernstein operators of max-product kind,, Fuzzy Set. Syst., 257 (2014), 41-66.  doi: 10.1016/j.fss.2013.04.010.

[13]

D. CostarelliA. R. Sambucini and G. Vinti, Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications,, Neural Comput. Appl., 31 (2019), 5069-5078.  doi: 10.1007/s00521-018-03998-6.

[14]

D. Costarelli and G. Vinti, Max-product neural network and quasi-interpolation operators activated by sigmoidal functions,, J. Approx. Theory, 209 (2016), 1-22.  doi: 10.1016/j.jat.2016.05.001.

[15]

S. G. Gal, Shape-Preserving Approximation by Real and Complex Polynomials, Birkhäuser, Boston-Basel-Berlin, 2008. doi: 10.1007/978-0-8176-4703-2.

[16]

T. Y. Gökçer and O. Duman, Approximation by max-min operators: A general theory and its applications,, Fuzzy Sets and Systems, 394 (2020), 146-161.  doi: 10.1016/j.fss.2019.11.007.

[17]

T. Y. Gökcer and O. Duman, Summation process by max-product operators,, Computational Analysis, 155 (2016), 59-67.  doi: 10.1007/978-3-319-28443-9_4.

[18]

S. Y. Güngör and N. Ispir, Approximation by Bernstein-Chlodowsky operators of max-product kind., Math. Commun., 23 (2018), 205-225. 

[19]

A. Holhoş, Weighted approximation of functions by Favard operators of max-product type,, Period. Math. Hungar., 77 (2018), 340-346.  doi: 10.1007/s10998-018-0249-9.

[20]

A. Holhoş, Weighted approximation of functions by Meyer-K önig and Zeller operators of max-product type,, Numer. Funct. Anal. Optim., 39 (2018), 689-703.  doi: 10.1080/01630563.2017.1413386.

[21]

S. Karakus and K. Demirci, Statistical $\sigma $-approximation to max-product operators,, Comput. Math. Appl., 61 (2011), 1024-1031.  doi: 10.1016/j.camwa.2010.12.052.

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