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.   Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[1]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[2]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[3]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[4]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[5]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

[6]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[7]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[8]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[9]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[10]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015

[11]

Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011

[12]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[13]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

 Impact Factor: 

Metrics

  • PDF downloads (52)
  • HTML views (204)
  • Cited by (0)

Other articles
by authors

[Back to Top]