May  2020, 3(2): 117-124. doi: 10.3934/mfc.2020009

Multivariate weighted kantorovich 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 Cluj-Napoca, Romania

* Corresponding author: Ana-Maria Acu

Received  December 2019 Revised  April 2020 Published  May 2020

Recently, some weighted Durrmeyer type operators were used as research tools in Learning Theory. In this paper we introduce a class of multidimensional weighted Kantorovich operators $ K_n $ on $ C(Q_d) $ where $ Q_d $ is the $ d $-dimensional hypercube $ [0,1]^d $. We show that each $ K_n $ has a unique invariant probability measure and determine this measure. Then, using results from approximation theory and the theory of ergodic operators, we find the limit of the iterates of $ K_n $ and give rates of convergence of the iterates toward the limit. Finally, we show that some Kantorovich type operators previously investigated in literature fall into the class of operators introduced in our paper. Other properties and applications, involving Learning Theory, will be presented in a forthcoming paper, where we will consider also operators on spaces of Lebesgue integrable functions on the hypercube $ Q_d $.

Citation: Ana-Maria Acu, Laura Hodis, Ioan Rasa. Multivariate weighted kantorovich operators. Mathematical Foundations of Computing, 2020, 3 (2) : 117-124. doi: 10.3934/mfc.2020009
References:
[1]

A.-M. Acu and H. Gonska, Classical Kantorovich operators revisited, Ukrainian Math., 71 (2019), 843-852. 

[2]

A.-M. Acu, N. Manav and D. F. Sofonea, Approximation properties of $\lambda$-Kantorovich operators, J. Inequal. Appl., (2018), Paper No. 202, 12 pp. doi: 10.1186/s13660-018-1795-7.

[3]

A.-M. Acu, M. Heilmann and I. Rasa, Iterates of convolution-type operators, submitted.

[4]

F. Altomare and M. Campiti, Korovkin-type approximation theory and its applications, in de Gruyter Studies in Mathematics, 17, Walter de Gruyter & Co., Berlin, (1994). doi: 10.1515/9783110884586.

[5]

F. AltomareM. C. MontanoV. Leonessa and I. Raşa, A generalization of Kantorovich operators for convex compact subsets, Banach J. Math. Anal., 11 (2017), 591-614.  doi: 10.1215/17358787-2017-0008.

[6]

F. AltomareM. C. MontanoV. Leonessa and I. Raşa, Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators, J. Math. Anal. Appl., 458 (2018), 153-173.  doi: 10.1016/j.jmaa.2017.08.034.

[7]

F. Altomare, M. C. Montano, V. Leonessa and I. Raşa, Markov operators, positive semigroups and approximation processes, in de Gruyter Studies in Mathematics, 61, De Gruyter, Berlin, (2014).

[8]

F. Altomare and I. Raşa, Lipschitz contractions, unique ergodicity and asymptotics of Markov semigroups, Boll. Unione Mat. Ital. (9), 5 (2012), 1-17. 

[9]

E. E. Berdysheva, M. Heilmann and K. Hennings, Pointwise convergence of the Bernstein-Durrmeyer operators with respect to a collection of measures, J. Approx. Theory, 251 (2020), 12 pp. doi: 10.1016/j.jat.2019.105339.

[10]

E. E. Berdysheva and K. Jetter, Multivariate Bernstein-Durrmeyer operators with arbitrary weight functions, J. Approx. Theory, 162 (2010), 576-598.  doi: 10.1016/j.jat.2009.11.005.

[11]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.

[12]

F. Cucker and D.-X. Zhou, Learning theory: An approximation theory viewpoint, Cambridge Monographs on Applied and Computational Mathematics, 24, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618796.

[13]

H. GonskaM. Heilmann and I. Raşa, Kantorovich operators of order $k$, Numer. Funct. Anal. Optim., 32 (2011), 717-738.  doi: 10.1080/01630563.2011.580877.

[14]

H. GonskaI. Raşa and M.-D. Rusu, Applications of an Ostrowski-type inequality, J. Comput. Anal. Appl., 14 (2012), 19-31. 

[15]

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

[16]

M. Heilmann and I. Raşa, $C_0$-semigroups associated with uniquely ergodic Kantorovich modifications of operators, Positivity, 22 (2018), 829-835.  doi: 10.1007/s11117-017-0547-0.

[17]

J. G. Kemeny and J. L. Snell, Finite Markov Chains, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976.

[18]

U. Krengel, Ergodic Theorems, De Gruyter Studies in Mathematics, 6, Walter de Gruyter & Co., Berlin, 1985. doi: 10.1515/9783110844641.

[19]

D. H. Mache and D. X. Zhou, Characterization theorems for the approximation by a family of operators, J. Approx. Theory, 84 (1996), 145-161.  doi: 10.1006/jath.1996.0012.

[20]

D.-X. Zhou, Converse theorems for multidimensional Kantorovich operators, Anal. Math., 19 (1993), 85-100.  doi: 10.1007/BF01904041.

[21]

D.-X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math., 25 (2006), 323-344.  doi: 10.1007/s10444-004-7206-2.

show all references

References:
[1]

A.-M. Acu and H. Gonska, Classical Kantorovich operators revisited, Ukrainian Math., 71 (2019), 843-852. 

[2]

A.-M. Acu, N. Manav and D. F. Sofonea, Approximation properties of $\lambda$-Kantorovich operators, J. Inequal. Appl., (2018), Paper No. 202, 12 pp. doi: 10.1186/s13660-018-1795-7.

[3]

A.-M. Acu, M. Heilmann and I. Rasa, Iterates of convolution-type operators, submitted.

[4]

F. Altomare and M. Campiti, Korovkin-type approximation theory and its applications, in de Gruyter Studies in Mathematics, 17, Walter de Gruyter & Co., Berlin, (1994). doi: 10.1515/9783110884586.

[5]

F. AltomareM. C. MontanoV. Leonessa and I. Raşa, A generalization of Kantorovich operators for convex compact subsets, Banach J. Math. Anal., 11 (2017), 591-614.  doi: 10.1215/17358787-2017-0008.

[6]

F. AltomareM. C. MontanoV. Leonessa and I. Raşa, Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators, J. Math. Anal. Appl., 458 (2018), 153-173.  doi: 10.1016/j.jmaa.2017.08.034.

[7]

F. Altomare, M. C. Montano, V. Leonessa and I. Raşa, Markov operators, positive semigroups and approximation processes, in de Gruyter Studies in Mathematics, 61, De Gruyter, Berlin, (2014).

[8]

F. Altomare and I. Raşa, Lipschitz contractions, unique ergodicity and asymptotics of Markov semigroups, Boll. Unione Mat. Ital. (9), 5 (2012), 1-17. 

[9]

E. E. Berdysheva, M. Heilmann and K. Hennings, Pointwise convergence of the Bernstein-Durrmeyer operators with respect to a collection of measures, J. Approx. Theory, 251 (2020), 12 pp. doi: 10.1016/j.jat.2019.105339.

[10]

E. E. Berdysheva and K. Jetter, Multivariate Bernstein-Durrmeyer operators with arbitrary weight functions, J. Approx. Theory, 162 (2010), 576-598.  doi: 10.1016/j.jat.2009.11.005.

[11]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.

[12]

F. Cucker and D.-X. Zhou, Learning theory: An approximation theory viewpoint, Cambridge Monographs on Applied and Computational Mathematics, 24, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618796.

[13]

H. GonskaM. Heilmann and I. Raşa, Kantorovich operators of order $k$, Numer. Funct. Anal. Optim., 32 (2011), 717-738.  doi: 10.1080/01630563.2011.580877.

[14]

H. GonskaI. Raşa and M.-D. Rusu, Applications of an Ostrowski-type inequality, J. Comput. Anal. Appl., 14 (2012), 19-31. 

[15]

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

[16]

M. Heilmann and I. Raşa, $C_0$-semigroups associated with uniquely ergodic Kantorovich modifications of operators, Positivity, 22 (2018), 829-835.  doi: 10.1007/s11117-017-0547-0.

[17]

J. G. Kemeny and J. L. Snell, Finite Markov Chains, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976.

[18]

U. Krengel, Ergodic Theorems, De Gruyter Studies in Mathematics, 6, Walter de Gruyter & Co., Berlin, 1985. doi: 10.1515/9783110844641.

[19]

D. H. Mache and D. X. Zhou, Characterization theorems for the approximation by a family of operators, J. Approx. Theory, 84 (1996), 145-161.  doi: 10.1006/jath.1996.0012.

[20]

D.-X. Zhou, Converse theorems for multidimensional Kantorovich operators, Anal. Math., 19 (1993), 85-100.  doi: 10.1007/BF01904041.

[21]

D.-X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math., 25 (2006), 323-344.  doi: 10.1007/s10444-004-7206-2.

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