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doi: 10.3934/mfc.2021026
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On bivariate Jain operators

Münüse Akçay and  Gülen Başcanbaz-Tunca ,

Ankara University, Faculty of Science, Department of Mathematics, Str. Dögol, 06100, Beşevler, Ankara, Turkey

* Corresponding author: Gülen Başcanbaz-Tunca

Received  June 2021 Revised  September 2021 Early access October 2021

In this paper we deal with bivariate extension of Jain operators. Using elementary method, we show that these opearators are non-increasing in $ n $ when the attached function is convex. Moreover, we demonstrate that these operators preserve the properties of modulus of continuity. Finally, we present a Voronovskaja type theorem for the sequence of bivariate Jain operators.

Keywords: Bivariate Jain operator, modulus of continuity, monotonicity, Voronovskaja type theorem.
Mathematics Subject Classification: Primary: 41A25, 41A36; Secondary: 41A63.
Citation: Münüse Akçay, Gülen Başcanbaz-Tunca. On bivariate Jain operators. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021026
References:
[1]

U. Abel and O. Agratini, Asymptotic behaviour of Jain operators, Numer Algor., 71 (2016), 553-565.  doi: 10.1007/s11075-015-0009-3.  Google Scholar

[2]

O. Agratini, Approximation properties of a class of linear operators, Math. Meth. Appl. Sci., 36 (2013), 2353-2358.  doi: 10.1002/mma.2758.  Google Scholar

[3]

O. Agratini, A stop over Jain operators and their generalizations, An. Univ. Vest Timiş. Ser. Mat.-Inform., 56 (2018), 28-42. doi: 10.2478/awutm-2018-0014.  Google Scholar

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C. Bardaro and I. Mantellini, On Pointwise approximation properties of multivariate Semi-discrete sampling type operators, Results Math., 72 (2017), 1449-1472.  doi: 10.1007/s00025-017-0667-7.  Google Scholar

[5]

F. Cao, C. Ding and Z. Xu, On multivariate Baskakov operator, J. Math. Anal. Appl., 307 (2005), 274-291. doi: 10.1016/j.jmaa.2004.10.061.  Google Scholar

[6]

N. Çetin and G. Başcanbaz-Tunca, Approximation by Jain-Schurer operators, Facta Univ. Ser. Math. Inform., 35 (2020), 1343-1356.  doi: 10.22190/fumi2005343c.  Google Scholar

[7]

E. W. Cheney and A. Sharma, Bernstein power series, Can. J. Math., 16 (1964), 241-252.  doi: 10.4153/CJM-1964-023-1.  Google Scholar

[8]

E. Deniz, Quantitative estimates for Jain-Kantorovich operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 65 (2016), 121-132.  doi: 10.1501/Commua1_0000000764.  Google Scholar

[9]

M. Dhamija and N. Deo, Jain–Durrmeyer operators associated with the inverse Pólya–Eggenberger distribution, Appl. Math. Comput., 286 (2016), 15-22.  doi: 10.1016/j.amc.2016.03.015.  Google Scholar

[10]

O. Doğru, R. N. Mohapatra and M. Örkcü, Approximation properties of genaralized Jain operators, Filomat, 30 (2016), 2359-2366.  doi: 10.2298/FIL1609359D.  Google Scholar

[11]

A. Farcaş, An asymptotic formula for Jain's operators, Stud. Univ. Babeş-Bolyai Math., 57 (2012), 511–517.  Google Scholar

[12]

A. Farcaş, An approximation property of the generalized Jain's operators of two variables, Math. Sci. Appl. E-Notes, 1 (2013), 158-164.   Google Scholar

[13]

V. Gupta, R. P. Agarwal and D. K. Verma, Approximation for a new sequence of summation-integral type operators, Adv. Math. Sci. Appl., 23 (2013), 35-42.   Google Scholar

[14]

V. Gupta and G. C. Greubel, Moment estimations of new Szász–Mirakyan–Durrmeyer operators, Appl. Math. Comput., 271 (2015), 540-547.  doi: 10.1016/j.amc.2015.09.037.  Google Scholar

[15]

G. C. Jain, Approximation of functions by a new class of linear operators, J. Austral. Math. Soc., 13 (1972), 271-276.  doi: 10.1017/S1446788700013689.  Google Scholar

[16]

Z. Li, Bernstein polynomials and modulus of continuity, J. Approx. Theory, 102 (2000), 171-174.  doi: 10.1006/jath.1999.3374.  Google Scholar

[17]

G. M. Mirakjan, Approximation of continuous functions with the aid of polynomials, Dokl. Akad. Nauk SSSR, 31 (1941), 201-205.   Google Scholar

[18]

A. Olgun, Some properties of the multivariate Szász operators, C. R. Acad. Bulg. Sci., 65 (2012), 139-146.   Google Scholar

[19]

A. Olgun, F. Taşdelen and A. Erençin, A generalization of Jain's operators, Appl. Math. Comput., 266 (2015), 6-11.  doi: 10.1016/j.amc.2015.05.060.  Google Scholar

[20]

M. A. Özarslan, Approximation properties of Jain-Stancu operators, Filomat, 30 (2016), 1081-1088.  doi: 10.2298/FIL1604081O.  Google Scholar

[21]

L. Rempulska and M. Skorupka, On Szasz-Mirakyan operators of functions of two variables, Le Matematiche, 53 (1998), 51-60.   Google Scholar

[22]

D. D. Stancu and E. I. Stoica, On the use Abel-Jensen type combinatorial formulas for construction and investigation of some algebraic polynomial operators of approximation, Stud. Univ. Babeş -Bolyai Math., 54 (2009), 167–182.  Google Scholar

[23]

O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Res. Natl. Bur. Stand., 45 (1950), 239-245.  doi: 10.6028/jres.045.024.  Google Scholar

[24]

S. Tarabie, On Jain-Beta linear operators, Appl. Math. Inf. Sci., 6 (2012), 213-216.   Google Scholar

[25]

S. Umar and Q. Razi, Approximation of function by a generalized Szász operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 34 (1985), 45-52.  doi: 10.1501/Commua1_0000000240.  Google Scholar

show all references

References:
[1]

U. Abel and O. Agratini, Asymptotic behaviour of Jain operators, Numer Algor., 71 (2016), 553-565.  doi: 10.1007/s11075-015-0009-3.  Google Scholar

[2]

O. Agratini, Approximation properties of a class of linear operators, Math. Meth. Appl. Sci., 36 (2013), 2353-2358.  doi: 10.1002/mma.2758.  Google Scholar

[3]

O. Agratini, A stop over Jain operators and their generalizations, An. Univ. Vest Timiş. Ser. Mat.-Inform., 56 (2018), 28-42. doi: 10.2478/awutm-2018-0014.  Google Scholar

[4]

C. Bardaro and I. Mantellini, On Pointwise approximation properties of multivariate Semi-discrete sampling type operators, Results Math., 72 (2017), 1449-1472.  doi: 10.1007/s00025-017-0667-7.  Google Scholar

[5]

F. Cao, C. Ding and Z. Xu, On multivariate Baskakov operator, J. Math. Anal. Appl., 307 (2005), 274-291. doi: 10.1016/j.jmaa.2004.10.061.  Google Scholar

[6]

N. Çetin and G. Başcanbaz-Tunca, Approximation by Jain-Schurer operators, Facta Univ. Ser. Math. Inform., 35 (2020), 1343-1356.  doi: 10.22190/fumi2005343c.  Google Scholar

[7]

E. W. Cheney and A. Sharma, Bernstein power series, Can. J. Math., 16 (1964), 241-252.  doi: 10.4153/CJM-1964-023-1.  Google Scholar

[8]

E. Deniz, Quantitative estimates for Jain-Kantorovich operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 65 (2016), 121-132.  doi: 10.1501/Commua1_0000000764.  Google Scholar

[9]

M. Dhamija and N. Deo, Jain–Durrmeyer operators associated with the inverse Pólya–Eggenberger distribution, Appl. Math. Comput., 286 (2016), 15-22.  doi: 10.1016/j.amc.2016.03.015.  Google Scholar

[10]

O. Doğru, R. N. Mohapatra and M. Örkcü, Approximation properties of genaralized Jain operators, Filomat, 30 (2016), 2359-2366.  doi: 10.2298/FIL1609359D.  Google Scholar

[11]

A. Farcaş, An asymptotic formula for Jain's operators, Stud. Univ. Babeş-Bolyai Math., 57 (2012), 511–517.  Google Scholar

[12]

A. Farcaş, An approximation property of the generalized Jain's operators of two variables, Math. Sci. Appl. E-Notes, 1 (2013), 158-164.   Google Scholar

[13]

V. Gupta, R. P. Agarwal and D. K. Verma, Approximation for a new sequence of summation-integral type operators, Adv. Math. Sci. Appl., 23 (2013), 35-42.   Google Scholar

[14]

V. Gupta and G. C. Greubel, Moment estimations of new Szász–Mirakyan–Durrmeyer operators, Appl. Math. Comput., 271 (2015), 540-547.  doi: 10.1016/j.amc.2015.09.037.  Google Scholar

[15]

G. C. Jain, Approximation of functions by a new class of linear operators, J. Austral. Math. Soc., 13 (1972), 271-276.  doi: 10.1017/S1446788700013689.  Google Scholar

[16]

Z. Li, Bernstein polynomials and modulus of continuity, J. Approx. Theory, 102 (2000), 171-174.  doi: 10.1006/jath.1999.3374.  Google Scholar

[17]

G. M. Mirakjan, Approximation of continuous functions with the aid of polynomials, Dokl. Akad. Nauk SSSR, 31 (1941), 201-205.   Google Scholar

[18]

A. Olgun, Some properties of the multivariate Szász operators, C. R. Acad. Bulg. Sci., 65 (2012), 139-146.   Google Scholar

[19]

A. Olgun, F. Taşdelen and A. Erençin, A generalization of Jain's operators, Appl. Math. Comput., 266 (2015), 6-11.  doi: 10.1016/j.amc.2015.05.060.  Google Scholar

[20]

M. A. Özarslan, Approximation properties of Jain-Stancu operators, Filomat, 30 (2016), 1081-1088.  doi: 10.2298/FIL1604081O.  Google Scholar

[21]

L. Rempulska and M. Skorupka, On Szasz-Mirakyan operators of functions of two variables, Le Matematiche, 53 (1998), 51-60.   Google Scholar

[22]

D. D. Stancu and E. I. Stoica, On the use Abel-Jensen type combinatorial formulas for construction and investigation of some algebraic polynomial operators of approximation, Stud. Univ. Babeş -Bolyai Math., 54 (2009), 167–182.  Google Scholar

[23]

O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Res. Natl. Bur. Stand., 45 (1950), 239-245.  doi: 10.6028/jres.045.024.  Google Scholar

[24]

S. Tarabie, On Jain-Beta linear operators, Appl. Math. Inf. Sci., 6 (2012), 213-216.   Google Scholar

[25]

S. Umar and Q. Razi, Approximation of function by a generalized Szász operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 34 (1985), 45-52.  doi: 10.1501/Commua1_0000000240.  Google Scholar

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