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June  2020, 28(2): 721-738. doi: 10.3934/era.2020037

## Approximation by multivariate Baskakov–Kantorovich operators in Orlicz spaces

 1 College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China 2 Department of Fundamental Courses, Zhengzhou University of Science and Technology, Zhengzhou 450064, Henan, China 3 Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan, China, School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China

* Corresponding author: Feng Qi

In honour of Professor Hari Mohan Srivastava on the occasion of his 80th birth anniversary

Received  January 2020 Revised  March 2020 Published  May 2020

Fund Project: The first author was supported by the Natural Science Foundation of Inner Mongolia (IMNSFC) under Grant No. 2016MS0118, China

Utilizing some properties of multivariate Baskakov–Kantorovich operators and using $K$-functional and a decomposition technique, the authors find two equivalent theorems between the $K$-functional and modulus of smoothness, and obtain a direct theorem in the Orlicz spaces.

Citation: 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
##### References:
 [1] A. M. Acu and V. Gupta, Direct results for certain summation-integral type Baskakov–Szász operators, Results Math., 72 (2017), 1161-1180.  doi: 10.1007/s00025-016-0603-2.  Google Scholar [2] P. N. Agrawal, V. Gupta and A. Sathish Kumar, Generalized Baskakov–Durrmeyer type operators, Rend. Circ. Mat. Palermo, 63 (2014), 193-209.  doi: 10.1007/s12215-014-0152-z.  Google Scholar [3] V. A. Baskakov, An instance of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 249–251. (Russian)  Google Scholar [4] F. L. Cao and Y. F. An, $L^p$ approximation by multivariate Baskakov–Durrmeyer operator, J. Inequal. Appl., 2011 (2011), Art. ID 158219, 7 pp. doi: 10.1155/2011/158219.  Google Scholar [5] F. L. Cao and C. M. Ding, $L^p$ approximation by multivariate Baskakov–Kantorovich operators, J. Math. Anal. Appl., 348 (2008), 856-861.  doi: 10.1016/j.jmaa.2008.05.049.  Google Scholar [6] F. L. Cao, C. M. Ding and Z. B. Xu, On multivariate Baskakov operators, J. Math. Anal. Appl., 307 (2005), 274-291.  doi: 10.1016/j.jmaa.2004.10.061.  Google Scholar [7] W. Chen and Z. Ditzian, Mixed and directional derivatives, Proc. Amer. Math. Soc., 108 (1990), 177-185.  doi: 10.1090/S0002-9939-1990-0994773-0.  Google Scholar [8] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Series in Computational Mathematics, 9. Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.  Google Scholar [9] A. Erençin, Durrmeyer type modification of generalized Baskakov operators, Appl. Math. Comput., 218 (2011), 4384-4390.  doi: 10.1016/j.amc.2011.10.014.  Google Scholar [10] I. Gadjev, Approximation of functions by Baskakov–Kantorovich operator, Results Math., 70 (2016), 385-400.  doi: 10.1007/s00025-016-0554-7.  Google Scholar [11] I. Gadjev, Strong converse result for Baskakov operator, Serdica Math. J., 40 (2014), 273-318.   Google Scholar [12] R. B. Gandhi, M. Deepmala and V. N. Mishra, Local and global results for modified Szász-Mirakjan operators, Math. Methods Appl. Sci., 40 (2017), 2491-2504.  doi: 10.1002/mma.4171.  Google Scholar [13] M. Goyal and P. N. Agrawal, Bézier variant of the generalized Baskakov Kantorovich operators, Boll. Unione Mat. Ital., 8 (2016), 229-238.  doi: 10.1007/s40574-015-0040-2.  Google Scholar [14] S. S. Guo, Q. L. Qi and G. F. Liu, The central approximation theorems for Baskakov–Bézier operators, J. Approx. Theory, 147 (2007), 112-124.  doi: 10.1016/j.jat.2005.02.010.  Google Scholar [15] V. Gupta and A. M. Acu, On Baskakov–Szász–Mirakyan-type operators preserving exponential type functions, Positivity, 22 (2018), 919-929.  doi: 10.1007/s11117-018-0553-x.  Google Scholar [16] F. Gürbüz, Generalized weighted Morrey estimates for Marcinkiewicz integrals with rough kernel associated with Schrödinger operator and their commutators, Chin. Ann. Math. Ser. B, 41 (2020), 77-98.  doi: 10.1007/s11401-019-0187-8.  Google Scholar [17] F. Gürbüz, Local campanato estimates for multilinear commutator operators with rough kernel on generalized local morrey spaces, J. Coupled Syst. Multiscale Dyn., 6 (2018), 71-79.  doi: 10.1166/jcsmd.2018.1143.  Google Scholar [18] F. Gürbüz, Some estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, Canad. Math. Bull., 60 (2017), 131-145.  doi: 10.4153/CMB-2016-067-8.  Google Scholar [19] F. Gürbüz, Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized Morrey spaces, Math. Notes, 101 (2017), 429-442.  doi: 10.1134/S0001434617030051.  Google Scholar [20] F. Gürbüz, Weighted Morrey and weighted fractional Sobolev-Morrey spaces estimates for a large class of pseudo-differential operators with smooth symbols, J. Pseudo-Differ. Oper. Appl., 7 (2016), 595-607.  doi: 10.1007/s11868-016-0158-8.  Google Scholar [21] L.-X. Han, B.-N. Guo and F. Qi, Equivalent theorem of approximation by linear combination of weighted Baskakov–Kantorovich operators in Orlicz spaces, J. Inequal Appl., 2019 (2019), 18 pp. doi: 10.1186/s13660-019-2174-8.  Google Scholar [22] L.-X. Han and F. Qi, On approximation by linear combinations of modified summation operators of integral type in Orlicz spaces, Mathematics, 7 (2019), Art. 6, 10 pp. doi: 10.3390/math7010006.  Google Scholar [23] L.-X. Han and G. Wu, Approximation by modified summation integral type operators in Orlicz spaces, Math. Appl. (Wuhan), 30 (2017), 613-622.   Google Scholar [24] L.-X. Han and G. Wu, Strong converse inequality of weighted simultaneous approximation for Gamma operators in Orlicz spaces $L^*_\Phi(0, \infty)$, Appl. Math. J. Chinese Univ. Ser. A, 31 (2016), 366–378. (Chinese)  Google Scholar [25] L.-X. Han, G. Wu and G. Liu, The equivalence of the smooth modulus with weights and a $K$-functional in Orlicz spaces and it's application, Acta Math. Sci. Ser. A, 34 (2014), 95–108. (Chinese)  Google Scholar [26] A. S. Kumar and T. Acar, Approximation by generalized Baskakov–Durrmeyer–Stancu type operators, Rend. Circ. Mat. Palermo, 65 (2016), 411-424.  doi: 10.1007/s12215-016-0242-1.  Google Scholar [27] V. N. Mishra, H. H. Khan, K. Khatri and L. N. Mishra, Hypergeometric representation for Baskakov–Durrmeyer–Stancu type operators, Bull. Math. Anal. Appl., 5 (2013), 18-26.   Google Scholar [28] V. N. Mishra, K. Khatri and L. N. Mishra, On simultaneous approximation for Baskakov–Durrmeyer–Stancu type operators, J. Ultra Scientist Phys. Sci. A, 24 (2012), 567-577.   Google Scholar [29] V. N. Mishra, K. Khatri and L. N. Mishra, Some approximation properties of $q$-Baskakov–Beta–Stancu type operators, Journal of Calculus of Variations, 2013 (2013), Art. ID 814824, 8 pp. doi: 10.1155/2013/814824.  Google Scholar [30] V. N. Mishra, K. Khatri, L. N. Mishra and Deepmala, Inverse result in simultaneous approximation by Baskakov–Durrmeyer–Stancu operators, J. Inequal. Appl., 2013 (2013), 11 pp. doi: 10.1186/1029-242X-2013-586.  Google Scholar [31] M. M. Rao and Z. D. Ren, Theory of Orlicz Space, Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, Inc., New York, 1991.  Google Scholar [32] V. Totik, An interpolation theorem and its applications to positive operators, Pacific J. Math., 111 (1984), 447-481.  doi: 10.2140/pjm.1984.111.447.  Google Scholar [33] V. Totik, Approximation by Bernstein polynomials, Amer. J. Math., 116 (1994), 995-1018.  doi: 10.2307/2375007.  Google Scholar [34] D. K. Verma, V. Gupta and P. N. Agrawal, Some approximation properties of Baskakov–Durrmeyer–Stancu operators, Appl. Math. Comput., 218 (2012), 6549-6556.  doi: 10.1016/j.amc.2011.12.031.  Google Scholar [35] C. G. Zhang and Z. H. Zhu, Preservation properties of the Baskakov–Kantorovich operators, Comput. Math. Appl., 57 (2009), 1450-1455.  doi: 10.1016/j.camwa.2009.01.027.  Google Scholar

show all references

##### References:
 [1] A. M. Acu and V. Gupta, Direct results for certain summation-integral type Baskakov–Szász operators, Results Math., 72 (2017), 1161-1180.  doi: 10.1007/s00025-016-0603-2.  Google Scholar [2] P. N. Agrawal, V. Gupta and A. Sathish Kumar, Generalized Baskakov–Durrmeyer type operators, Rend. Circ. Mat. Palermo, 63 (2014), 193-209.  doi: 10.1007/s12215-014-0152-z.  Google Scholar [3] V. A. Baskakov, An instance of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 249–251. (Russian)  Google Scholar [4] F. L. Cao and Y. F. An, $L^p$ approximation by multivariate Baskakov–Durrmeyer operator, J. Inequal. Appl., 2011 (2011), Art. ID 158219, 7 pp. doi: 10.1155/2011/158219.  Google Scholar [5] F. L. Cao and C. M. Ding, $L^p$ approximation by multivariate Baskakov–Kantorovich operators, J. Math. Anal. Appl., 348 (2008), 856-861.  doi: 10.1016/j.jmaa.2008.05.049.  Google Scholar [6] F. L. Cao, C. M. Ding and Z. B. Xu, On multivariate Baskakov operators, J. Math. Anal. Appl., 307 (2005), 274-291.  doi: 10.1016/j.jmaa.2004.10.061.  Google Scholar [7] W. Chen and Z. Ditzian, Mixed and directional derivatives, Proc. Amer. Math. Soc., 108 (1990), 177-185.  doi: 10.1090/S0002-9939-1990-0994773-0.  Google Scholar [8] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Series in Computational Mathematics, 9. Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.  Google Scholar [9] A. Erençin, Durrmeyer type modification of generalized Baskakov operators, Appl. Math. Comput., 218 (2011), 4384-4390.  doi: 10.1016/j.amc.2011.10.014.  Google Scholar [10] I. Gadjev, Approximation of functions by Baskakov–Kantorovich operator, Results Math., 70 (2016), 385-400.  doi: 10.1007/s00025-016-0554-7.  Google Scholar [11] I. Gadjev, Strong converse result for Baskakov operator, Serdica Math. J., 40 (2014), 273-318.   Google Scholar [12] R. B. Gandhi, M. Deepmala and V. N. Mishra, Local and global results for modified Szász-Mirakjan operators, Math. Methods Appl. Sci., 40 (2017), 2491-2504.  doi: 10.1002/mma.4171.  Google Scholar [13] M. Goyal and P. N. Agrawal, Bézier variant of the generalized Baskakov Kantorovich operators, Boll. Unione Mat. Ital., 8 (2016), 229-238.  doi: 10.1007/s40574-015-0040-2.  Google Scholar [14] S. S. Guo, Q. L. Qi and G. F. Liu, The central approximation theorems for Baskakov–Bézier operators, J. Approx. Theory, 147 (2007), 112-124.  doi: 10.1016/j.jat.2005.02.010.  Google Scholar [15] V. Gupta and A. M. Acu, On Baskakov–Szász–Mirakyan-type operators preserving exponential type functions, Positivity, 22 (2018), 919-929.  doi: 10.1007/s11117-018-0553-x.  Google Scholar [16] F. Gürbüz, Generalized weighted Morrey estimates for Marcinkiewicz integrals with rough kernel associated with Schrödinger operator and their commutators, Chin. Ann. Math. Ser. B, 41 (2020), 77-98.  doi: 10.1007/s11401-019-0187-8.  Google Scholar [17] F. Gürbüz, Local campanato estimates for multilinear commutator operators with rough kernel on generalized local morrey spaces, J. Coupled Syst. Multiscale Dyn., 6 (2018), 71-79.  doi: 10.1166/jcsmd.2018.1143.  Google Scholar [18] F. Gürbüz, Some estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, Canad. Math. Bull., 60 (2017), 131-145.  doi: 10.4153/CMB-2016-067-8.  Google Scholar [19] F. Gürbüz, Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized Morrey spaces, Math. Notes, 101 (2017), 429-442.  doi: 10.1134/S0001434617030051.  Google Scholar [20] F. Gürbüz, Weighted Morrey and weighted fractional Sobolev-Morrey spaces estimates for a large class of pseudo-differential operators with smooth symbols, J. Pseudo-Differ. Oper. Appl., 7 (2016), 595-607.  doi: 10.1007/s11868-016-0158-8.  Google Scholar [21] L.-X. Han, B.-N. Guo and F. Qi, Equivalent theorem of approximation by linear combination of weighted Baskakov–Kantorovich operators in Orlicz spaces, J. Inequal Appl., 2019 (2019), 18 pp. doi: 10.1186/s13660-019-2174-8.  Google Scholar [22] L.-X. Han and F. Qi, On approximation by linear combinations of modified summation operators of integral type in Orlicz spaces, Mathematics, 7 (2019), Art. 6, 10 pp. doi: 10.3390/math7010006.  Google Scholar [23] L.-X. Han and G. Wu, Approximation by modified summation integral type operators in Orlicz spaces, Math. Appl. (Wuhan), 30 (2017), 613-622.   Google Scholar [24] L.-X. Han and G. Wu, Strong converse inequality of weighted simultaneous approximation for Gamma operators in Orlicz spaces $L^*_\Phi(0, \infty)$, Appl. Math. J. Chinese Univ. Ser. A, 31 (2016), 366–378. (Chinese)  Google Scholar [25] L.-X. Han, G. Wu and G. Liu, The equivalence of the smooth modulus with weights and a $K$-functional in Orlicz spaces and it's application, Acta Math. Sci. Ser. A, 34 (2014), 95–108. (Chinese)  Google Scholar [26] A. S. Kumar and T. Acar, Approximation by generalized Baskakov–Durrmeyer–Stancu type operators, Rend. Circ. Mat. Palermo, 65 (2016), 411-424.  doi: 10.1007/s12215-016-0242-1.  Google Scholar [27] V. N. Mishra, H. H. Khan, K. Khatri and L. N. Mishra, Hypergeometric representation for Baskakov–Durrmeyer–Stancu type operators, Bull. Math. Anal. Appl., 5 (2013), 18-26.   Google Scholar [28] V. N. Mishra, K. Khatri and L. N. Mishra, On simultaneous approximation for Baskakov–Durrmeyer–Stancu type operators, J. Ultra Scientist Phys. Sci. A, 24 (2012), 567-577.   Google Scholar [29] V. N. Mishra, K. Khatri and L. N. Mishra, Some approximation properties of $q$-Baskakov–Beta–Stancu type operators, Journal of Calculus of Variations, 2013 (2013), Art. ID 814824, 8 pp. doi: 10.1155/2013/814824.  Google Scholar [30] V. N. Mishra, K. Khatri, L. N. Mishra and Deepmala, Inverse result in simultaneous approximation by Baskakov–Durrmeyer–Stancu operators, J. Inequal. Appl., 2013 (2013), 11 pp. doi: 10.1186/1029-242X-2013-586.  Google Scholar [31] M. M. Rao and Z. D. Ren, Theory of Orlicz Space, Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, Inc., New York, 1991.  Google Scholar [32] V. Totik, An interpolation theorem and its applications to positive operators, Pacific J. Math., 111 (1984), 447-481.  doi: 10.2140/pjm.1984.111.447.  Google Scholar [33] V. Totik, Approximation by Bernstein polynomials, Amer. J. Math., 116 (1994), 995-1018.  doi: 10.2307/2375007.  Google Scholar [34] D. K. Verma, V. Gupta and P. N. Agrawal, Some approximation properties of Baskakov–Durrmeyer–Stancu operators, Appl. Math. Comput., 218 (2012), 6549-6556.  doi: 10.1016/j.amc.2011.12.031.  Google Scholar [35] C. G. Zhang and Z. H. Zhu, Preservation properties of the Baskakov–Kantorovich operators, Comput. Math. Appl., 57 (2009), 1450-1455.  doi: 10.1016/j.camwa.2009.01.027.  Google Scholar
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