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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. AgrawalV. 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. CaoC. 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. GandhiM. 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. GuoQ. 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. MishraH. H. KhanK. 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. MishraK. 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. VermaV. 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. AgrawalV. 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. CaoC. 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. GandhiM. 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. GuoQ. 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. MishraH. H. KhanK. 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. MishraK. 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. VermaV. 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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