# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2021042
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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:

show all references

##### References:
 [1] 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 [2] Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008 [3] Ling-Xiong Han, Wen-Hui Li, Feng Qi. Approximation by multivariate Baskakov–Kantorovich operators in Orlicz spaces. Electronic Research Archive, 2020, 28 (2) : 721-738. doi: 10.3934/era.2020037 [4] Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5507-5519. doi: 10.3934/dcds.2013.33.5507 [5] David Gómez-Ullate, Niky Kamran, Robert Milson. Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 85-106. doi: 10.3934/dcds.2007.18.85 [6] Uğur Kadak, Faruk Özger. A numerical comparative study of generalized Bernstein-Kantorovich operators. Mathematical Foundations of Computing, 2021, 4 (4) : 311-332. doi: 10.3934/mfc.2021021 [7] Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223 [8] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [9] Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811 [10] Lucian Coroianu, Danilo Costarelli, Sorin G. Gal, Gianluca Vinti. Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4213-4225. doi: 10.3934/cpaa.2020189 [11] Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73 [12] Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101 [13] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085 [14] Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123 [15] Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140 [16] Dong Ye, Feng Zhou. Invariant criteria for existence of bounded positive solutions. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 413-424. doi: 10.3934/dcds.2005.12.413 [17] Amir Mohammadi. Measures invariant under horospherical subgroups in positive characteristic. Journal of Modern Dynamics, 2011, 5 (2) : 237-254. doi: 10.3934/jmd.2011.5.237 [18] Giuseppina di Blasio, Filomena Feo, Maria Rosaria Posteraro. Existence results for nonlinear elliptic equations related to Gauss measure in a limit case. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1497-1506. doi: 10.3934/cpaa.2008.7.1497 [19] Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054 [20] Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435