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A new kind of Bi-variate $ \lambda $ -Bernstein-Kantorovich type operator with shifted knots and its associated GBS form
On bivariate Jain operators
Ankara University, Faculty of Science, Department of Mathematics, Str. Dögol, 06100, Beşevler, Ankara, Turkey |
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.
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. |
| [2] |
O. Agratini,
Approximation properties of a class of linear operators, Math. Meth. Appl. Sci., 36 (2013), 2353-2358.
doi: 10.1002/mma.2758. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
E. W. Cheney and A. Sharma,
Bernstein power series, Can. J. Math., 16 (1964), 241-252.
doi: 10.4153/CJM-1964-023-1. |
| [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. |
| [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. |
| [10] |
O. Doğru, R. N. Mohapatra and M. Örkcü,
Approximation properties of genaralized Jain operators, Filomat, 30 (2016), 2359-2366.
doi: 10.2298/FIL1609359D. |
| [11] |
A. Farcaş, An asymptotic formula for Jain's operators, Stud. Univ. Babeş-Bolyai Math., 57 (2012), 511–517. |
| [12] |
A. Farcaş,
An approximation property of the generalized Jain's operators of two variables, Math. Sci. Appl. E-Notes, 1 (2013), 158-164.
|
| [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.
|
| [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. |
| [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. |
| [16] |
Z. Li,
Bernstein polynomials and modulus of continuity, J. Approx. Theory, 102 (2000), 171-174.
doi: 10.1006/jath.1999.3374. |
| [17] |
G. M. Mirakjan,
Approximation of continuous functions with the aid of polynomials, Dokl. Akad. Nauk SSSR, 31 (1941), 201-205.
|
| [18] |
A. Olgun,
Some properties of the multivariate Szász operators, C. R. Acad. Bulg. Sci., 65 (2012), 139-146.
|
| [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. |
| [20] |
M. A. Özarslan,
Approximation properties of Jain-Stancu operators, Filomat, 30 (2016), 1081-1088.
doi: 10.2298/FIL1604081O. |
| [21] |
L. Rempulska and M. Skorupka,
On Szasz-Mirakyan operators of functions of two variables, Le Matematiche, 53 (1998), 51-60.
|
| [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. |
| [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. |
| [24] |
S. Tarabie,
On Jain-Beta linear operators, Appl. Math. Inf. Sci., 6 (2012), 213-216.
|
| [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. |
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. |
| [2] |
O. Agratini,
Approximation properties of a class of linear operators, Math. Meth. Appl. Sci., 36 (2013), 2353-2358.
doi: 10.1002/mma.2758. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
E. W. Cheney and A. Sharma,
Bernstein power series, Can. J. Math., 16 (1964), 241-252.
doi: 10.4153/CJM-1964-023-1. |
| [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. |
| [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. |
| [10] |
O. Doğru, R. N. Mohapatra and M. Örkcü,
Approximation properties of genaralized Jain operators, Filomat, 30 (2016), 2359-2366.
doi: 10.2298/FIL1609359D. |
| [11] |
A. Farcaş, An asymptotic formula for Jain's operators, Stud. Univ. Babeş-Bolyai Math., 57 (2012), 511–517. |
| [12] |
A. Farcaş,
An approximation property of the generalized Jain's operators of two variables, Math. Sci. Appl. E-Notes, 1 (2013), 158-164.
|
| [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.
|
| [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. |
| [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. |
| [16] |
Z. Li,
Bernstein polynomials and modulus of continuity, J. Approx. Theory, 102 (2000), 171-174.
doi: 10.1006/jath.1999.3374. |
| [17] |
G. M. Mirakjan,
Approximation of continuous functions with the aid of polynomials, Dokl. Akad. Nauk SSSR, 31 (1941), 201-205.
|
| [18] |
A. Olgun,
Some properties of the multivariate Szász operators, C. R. Acad. Bulg. Sci., 65 (2012), 139-146.
|
| [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. |
| [20] |
M. A. Özarslan,
Approximation properties of Jain-Stancu operators, Filomat, 30 (2016), 1081-1088.
doi: 10.2298/FIL1604081O. |
| [21] |
L. Rempulska and M. Skorupka,
On Szasz-Mirakyan operators of functions of two variables, Le Matematiche, 53 (1998), 51-60.
|
| [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. |
| [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. |
| [24] |
S. Tarabie,
On Jain-Beta linear operators, Appl. Math. Inf. Sci., 6 (2012), 213-216.
|
| [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. |
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