August  2022, 5(3): 157-172. doi: 10.3934/mfc.2021025

A new kind of Bi-variate $ \lambda $ -Bernstein-Kantorovich type operator with shifted knots and its associated GBS form

1. 

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee - 247667, India

2. 

Department of Mathematics, University of Prishtina, Prishtina, Kosovo

* Corresponding author: Rahul Shukla

Dedicated to Prof. R. P. Agarwal on his 74th birthday

Received  June 2021 Revised  August 2021 Published  August 2022 Early access  October 2021

Fund Project: The third author is supported by MoE, Govt. of India as a Senior Research Fellow

In this paper, we introduce a bi-variate case of a new kind of $ \lambda $-Bernstein-Kantorovich type operator with shifted knots defined by Rahman et al. [31]. The rate of convergence of the bi-variate operators is obtained in terms of the complete and partial moduli of continuity. Next, we give an error estimate in the approximation of a function in the Lipschitz class and establish a Voronovskaja type theorem. Also, we define the associated GBS(Generalized Boolean Sum) operators and study the degree of approximation of Bögel continuous and Bögel differentiable functions by these operators with the aid of the mixed modulus of smoothness. Finally, we show the rate of convergence of the bi-variate operators and their GBS case for certain functions by illustrative graphics and tables using MATLAB algorithms.

Citation: Purshottam Narain Agrawal, Behar Baxhaku, Rahul Shukla. A new kind of Bi-variate $ \lambda $ -Bernstein-Kantorovich type operator with shifted knots and its associated GBS form. Mathematical Foundations of Computing, 2022, 5 (3) : 157-172. doi: 10.3934/mfc.2021025
References:
[1]

A. M. Acu, N. Manav and D. F. Sofonea, Approximation properties of $\lambda$-Kantorovich operators, J. Inequal. Appl., 202 (2018), Paper No. 202, 12 pp. doi: 10.1186/s13660-018-1795-7.

[2]

P. N. AgrawalB. Baxhaku and R. Shukla, On q-analogue of a parametric generalization of Baskakov operators, Math. Methods Appl. Sci., 44 (2021), 5989-6004.  doi: 10.1002/mma.7163.

[3]

R. P. Agarwal and V. Gupta, On $q$-analogue of a complex summation-integral type operators in compact disk, J. Inequal. Appl., 2012 (2012), Article number: 111. doi: 10.1186/1029-242X-2012-111.

[4]

P. N. Agrawal and N. Ispir, Degree of approximation for bivariate Chlodowsky-Szász-Charlier type operators, Results Math., 69 (2016), 365-385.  doi: 10.1007/s00025-015-0495-6.

[5]

P. N. AgrawalN. Ispir and A. Kajla, GBS operators of Lupaş-Durrmeyer type based on Pólya distribution, Results Math., 69 (2016), 397-418.  doi: 10.1007/s00025-015-0507-6.

[6]

G. A. Anastassiou and S. G. Gal, Approximation Theory: Moduli of Continuity and Global Smoothness Preservation, Birkhäuser Boston, Inc., Boston, MA, 2000. doi: 10.1007/978-1-4612-1360-4.

[7]

A. Aral and V. Gupta, On the Durrmeyer type modification of the $q$-Baskakov type operators, Nonlinear Anal., 72 (2010), 1171-1180.  doi: 10.1016/j.na.2009.07.052.

[8]

C. BadeaI. BadeaC. Cottin and H. H. Gonska, Notes on degree of approximation of B-continuous and B-differentiable functions, Approx. Theory Appl., 4 (1988), 95-108. 

[9]

C. BadeaI. Badea and H. H. Gonska, A test function theorem and approximation by pseudopolynomials, Bull. Aust. Math. Soc., 34 (1986), 53-64.  doi: 10.1017/S0004972700004494.

[10]

C. Badea and C. Cottin, Korovkin-type theorem for generalized boolean sum operators, In Colloq. Math. Soc. Jńos Bolyai, North-Holland, Amsterdam, 58 (1991), 51-67. 

[11]

D. Bărbosu, Kantorovich-Schurer bivariate operators, Miskolc Math. Notes, 5 (2004), 129-136.  doi: 10.18514/MMN.2004.71.

[12]

D. BărbosuA.-M. Acu and C. V. Muraru, On certain GBS-Durrmeyer operators based on $q$-integers, Turk. J. Math., 41 (2017), 368-380.  doi: 10.3906/mat-1601-34.

[13]

D. Bărbosu and C. V. Muraru, Approximating B-continuous functions using GBS operators of Bernstein-Schurer-Stancu type based on $q$-integers, Appl. Math. Comput., 259 (2015), 80-87.  doi: 10.1016/j.amc.2015.02.030.

[14]

D. Bărbosu and O. T. Pop, A note on the GBS Bernstein's approximation formula, Annals Univ. of Craiova, Math. Comp. Sci. Ser., 35 (2008), 1-6. 

[15]

I. BârsanP. Braica and M. Fǎrcaş, About approximation of B-continuous functions of several variables by generalized boolean sum operators of Bernstein type on a simplex, Creat. Math. Inform., 20 (2011), 20-23.  doi: 10.37193/CMI.2011.01.03.

[16]

B. Baxhaku and P. N. Agrawal, Degree of approximation for bivariate extension of Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators, Appl. Math. Comput., 306 (2017), 56-72.  doi: 10.1016/j.amc.2017.02.007.

[17]

B. Baxhaku, P. N. Agrawal and R. Shukla, Bivariate positive linear operators constructed by means of $q$-Lagrange polynomials, J. Math. Anal. Appl., 491 (2020), 124337, 24 pp. doi: 10.1016/j.jmaa.2020.124337.

[18]

Q. B. Cai, B.-Y. Lian and G. Zhou, Approximation properties of $\lambda$-Bernstein operators, J. Inequal. Appl., 2018 (2018), Paper No. 61, 11 pp. doi: 10.1186/s13660-018-1653-7.

[19]

Q. B. Cai and G. Zhou, Blending type approximation by GBS operators of bivariate tensor product of $\lambda$-Bernstein-Kantorovich type, J. Inequal. Appl., 2018 (2018), Paper No. 268, 11 pp. doi: 10.1186/s13660-018-1862-0.

[20]

Q. B. Cai and G. Zhou, On $(p, q)$-analogue of Kantorovich type Bernstein-Stancu-Schurer operators, Appl. Math. Comput., 276 (2016), 12-20.  doi: 10.1016/j.amc.2015.12.006.

[21]

D. Cárdenas-Morales and V. Gupta, Two families of Bernstein-Durrmeyer type operators, Appl. Math. Comput., 248 (2014), 342-353.  doi: 10.1016/j.amc.2014.09.094.

[22]

C. Cottin, Mixed K-functionals: A measure of smoothness for blending-type approximation, Math. Z., 204 (1990), 69-83.  doi: 10.1007/BF02570860.

[23]

E. Dobrescu and I. Matei, The approximation by Bernstein type polynomials of bidimensional continuous functions, An. Univ. Timişoara Ser. Şti. Mat.-Fiz., 4 (1966), 85-90. 

[24]

O. Doğru and K. Kanat, On statistical approximation properties of the Kantorovich type Lupaş operators, Math. Comput. Modelling, 55 (2012), 1610-1621.  doi: 10.1016/j.mcm.2011.10.059.

[25]

M. D. Farcas, About approximation of B-continuous and B-differentiable functions of three variables by GBS operators of Bernstein type, Creat. Math. Inform., 17 (2008), 20-27. 

[26]

A. D. Gadjiev and A. M. Ghorbanalizadeh, Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables, Appl. Math. Comput., 216 (2010), 890-901.  doi: 10.1016/j.amc.2010.01.099.

[27]

N. K. GovilV. Gupta and D. Soybaş, Certain new class of Durrmeyer type operators, Appl. Math. Comput., 225 (2013), 195-203.  doi: 10.1016/j.amc.2013.09.030.

[28]

V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, Berlin, 2014. doi: 10.1007/978-3-319-02765-4.

[29]

V. Gupta, T. M. Rassias, P. N. Agrawal and A. M. Acu, Recent Advances in Constructive Approximation Theory, Series: Springer Optimization and its Applications, Springer, Cham, 2018. doi: 10.1007/978-3-319-92165-5.

[30]

M. Örkcü and O. Doğru, Weighted statistical approximation by Kantorovich type $q$-Szász-Mirakjan operators, Appl. Math. Comput., 217 (2011), 7913-7919.  doi: 10.1016/j.amc.2011.03.009.

[31]

S. Rahman, M. Mursaleen and A. M. Acu, Approximation properties of $\lambda$-Bernstein-Kantorovich operators with shifted knots, Math. Methods Appl. Sci., (2019), 4042–4053. doi: 10.1002/mma.5632.

[32]

V. I. Volkov, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, Dokl. Akad. Nauk. SSSR (N.S), 115 (1957), 17-19. 

show all references

References:
[1]

A. M. Acu, N. Manav and D. F. Sofonea, Approximation properties of $\lambda$-Kantorovich operators, J. Inequal. Appl., 202 (2018), Paper No. 202, 12 pp. doi: 10.1186/s13660-018-1795-7.

[2]

P. N. AgrawalB. Baxhaku and R. Shukla, On q-analogue of a parametric generalization of Baskakov operators, Math. Methods Appl. Sci., 44 (2021), 5989-6004.  doi: 10.1002/mma.7163.

[3]

R. P. Agarwal and V. Gupta, On $q$-analogue of a complex summation-integral type operators in compact disk, J. Inequal. Appl., 2012 (2012), Article number: 111. doi: 10.1186/1029-242X-2012-111.

[4]

P. N. Agrawal and N. Ispir, Degree of approximation for bivariate Chlodowsky-Szász-Charlier type operators, Results Math., 69 (2016), 365-385.  doi: 10.1007/s00025-015-0495-6.

[5]

P. N. AgrawalN. Ispir and A. Kajla, GBS operators of Lupaş-Durrmeyer type based on Pólya distribution, Results Math., 69 (2016), 397-418.  doi: 10.1007/s00025-015-0507-6.

[6]

G. A. Anastassiou and S. G. Gal, Approximation Theory: Moduli of Continuity and Global Smoothness Preservation, Birkhäuser Boston, Inc., Boston, MA, 2000. doi: 10.1007/978-1-4612-1360-4.

[7]

A. Aral and V. Gupta, On the Durrmeyer type modification of the $q$-Baskakov type operators, Nonlinear Anal., 72 (2010), 1171-1180.  doi: 10.1016/j.na.2009.07.052.

[8]

C. BadeaI. BadeaC. Cottin and H. H. Gonska, Notes on degree of approximation of B-continuous and B-differentiable functions, Approx. Theory Appl., 4 (1988), 95-108. 

[9]

C. BadeaI. Badea and H. H. Gonska, A test function theorem and approximation by pseudopolynomials, Bull. Aust. Math. Soc., 34 (1986), 53-64.  doi: 10.1017/S0004972700004494.

[10]

C. Badea and C. Cottin, Korovkin-type theorem for generalized boolean sum operators, In Colloq. Math. Soc. Jńos Bolyai, North-Holland, Amsterdam, 58 (1991), 51-67. 

[11]

D. Bărbosu, Kantorovich-Schurer bivariate operators, Miskolc Math. Notes, 5 (2004), 129-136.  doi: 10.18514/MMN.2004.71.

[12]

D. BărbosuA.-M. Acu and C. V. Muraru, On certain GBS-Durrmeyer operators based on $q$-integers, Turk. J. Math., 41 (2017), 368-380.  doi: 10.3906/mat-1601-34.

[13]

D. Bărbosu and C. V. Muraru, Approximating B-continuous functions using GBS operators of Bernstein-Schurer-Stancu type based on $q$-integers, Appl. Math. Comput., 259 (2015), 80-87.  doi: 10.1016/j.amc.2015.02.030.

[14]

D. Bărbosu and O. T. Pop, A note on the GBS Bernstein's approximation formula, Annals Univ. of Craiova, Math. Comp. Sci. Ser., 35 (2008), 1-6. 

[15]

I. BârsanP. Braica and M. Fǎrcaş, About approximation of B-continuous functions of several variables by generalized boolean sum operators of Bernstein type on a simplex, Creat. Math. Inform., 20 (2011), 20-23.  doi: 10.37193/CMI.2011.01.03.

[16]

B. Baxhaku and P. N. Agrawal, Degree of approximation for bivariate extension of Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators, Appl. Math. Comput., 306 (2017), 56-72.  doi: 10.1016/j.amc.2017.02.007.

[17]

B. Baxhaku, P. N. Agrawal and R. Shukla, Bivariate positive linear operators constructed by means of $q$-Lagrange polynomials, J. Math. Anal. Appl., 491 (2020), 124337, 24 pp. doi: 10.1016/j.jmaa.2020.124337.

[18]

Q. B. Cai, B.-Y. Lian and G. Zhou, Approximation properties of $\lambda$-Bernstein operators, J. Inequal. Appl., 2018 (2018), Paper No. 61, 11 pp. doi: 10.1186/s13660-018-1653-7.

[19]

Q. B. Cai and G. Zhou, Blending type approximation by GBS operators of bivariate tensor product of $\lambda$-Bernstein-Kantorovich type, J. Inequal. Appl., 2018 (2018), Paper No. 268, 11 pp. doi: 10.1186/s13660-018-1862-0.

[20]

Q. B. Cai and G. Zhou, On $(p, q)$-analogue of Kantorovich type Bernstein-Stancu-Schurer operators, Appl. Math. Comput., 276 (2016), 12-20.  doi: 10.1016/j.amc.2015.12.006.

[21]

D. Cárdenas-Morales and V. Gupta, Two families of Bernstein-Durrmeyer type operators, Appl. Math. Comput., 248 (2014), 342-353.  doi: 10.1016/j.amc.2014.09.094.

[22]

C. Cottin, Mixed K-functionals: A measure of smoothness for blending-type approximation, Math. Z., 204 (1990), 69-83.  doi: 10.1007/BF02570860.

[23]

E. Dobrescu and I. Matei, The approximation by Bernstein type polynomials of bidimensional continuous functions, An. Univ. Timişoara Ser. Şti. Mat.-Fiz., 4 (1966), 85-90. 

[24]

O. Doğru and K. Kanat, On statistical approximation properties of the Kantorovich type Lupaş operators, Math. Comput. Modelling, 55 (2012), 1610-1621.  doi: 10.1016/j.mcm.2011.10.059.

[25]

M. D. Farcas, About approximation of B-continuous and B-differentiable functions of three variables by GBS operators of Bernstein type, Creat. Math. Inform., 17 (2008), 20-27. 

[26]

A. D. Gadjiev and A. M. Ghorbanalizadeh, Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables, Appl. Math. Comput., 216 (2010), 890-901.  doi: 10.1016/j.amc.2010.01.099.

[27]

N. K. GovilV. Gupta and D. Soybaş, Certain new class of Durrmeyer type operators, Appl. Math. Comput., 225 (2013), 195-203.  doi: 10.1016/j.amc.2013.09.030.

[28]

V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, Berlin, 2014. doi: 10.1007/978-3-319-02765-4.

[29]

V. Gupta, T. M. Rassias, P. N. Agrawal and A. M. Acu, Recent Advances in Constructive Approximation Theory, Series: Springer Optimization and its Applications, Springer, Cham, 2018. doi: 10.1007/978-3-319-92165-5.

[30]

M. Örkcü and O. Doğru, Weighted statistical approximation by Kantorovich type $q$-Szász-Mirakjan operators, Appl. Math. Comput., 217 (2011), 7913-7919.  doi: 10.1016/j.amc.2011.03.009.

[31]

S. Rahman, M. Mursaleen and A. M. Acu, Approximation properties of $\lambda$-Bernstein-Kantorovich operators with shifted knots, Math. Methods Appl. Sci., (2019), 4042–4053. doi: 10.1002/mma.5632.

[32]

V. I. Volkov, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, Dokl. Akad. Nauk. SSSR (N.S), 115 (1957), 17-19. 

Figure 1.  The convergence of operators $ K_{m_{1}, m_{2}, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(f(t, s);x, y) $ to the function $ f(x, y). $
Figure 2.  The convergence of operators $ T_{m_{1}, m_{2}, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(.;x, y) $ to the function $ f(x, y). $
Table 1.  Error of approximation for operators $ K_{m_{1}, m_{2}, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(f(t, s);x, y) $ to the function $ f(x, y). $
(x, y) $ |K_{5, 5, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(f)-f(x, y)| $ $ |K_{20, 20, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(f)-f(x, y)| $ $ |K_{30, 30, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(f)-f(x, y)| $
(0, 0.1) 0.0014 0.000046 0.0000017
(0, 0.2) 0.0023 0.000113 0.0000469
(0, 0.4) 0.0042 0.000328 0.0001480
(0, 0.6) 0.0065 0.000649 0.0003044
(0.3, 0.2) 0.0104 0.002238 0.0014384
(0.4, 0.6) 0.0574 0.005843 0.0040092
(0.7, 0.85) 0.0044 0.015745 0.0010459
(0.9, 0.9) 0.0041 0.049901 0.0343900
(x, y) $ |K_{5, 5, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(f)-f(x, y)| $ $ |K_{20, 20, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(f)-f(x, y)| $ $ |K_{30, 30, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(f)-f(x, y)| $
(0, 0.1) 0.0014 0.000046 0.0000017
(0, 0.2) 0.0023 0.000113 0.0000469
(0, 0.4) 0.0042 0.000328 0.0001480
(0, 0.6) 0.0065 0.000649 0.0003044
(0.3, 0.2) 0.0104 0.002238 0.0014384
(0.4, 0.6) 0.0574 0.005843 0.0040092
(0.7, 0.85) 0.0044 0.015745 0.0010459
(0.9, 0.9) 0.0041 0.049901 0.0343900
Table 2.  Error of approximation for operators $ T_{m_{1}, m_{2}, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}(.;x, y) $ to the function $ f(x, y). $
(x, y) $ |T_{5, 5, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}-f(x, y)| $ $ |T_{20, 20, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}-f(x, y)| $ $ |T_{30, 30, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}-f(x, y)| $
(0, 0.1) 0.000347569 0.0000078 0.00000246
(0, 0.2) 0.000201932 0.0000009 0.00000039
(0, 0.4) 0.000862538 0.0000596 0.00002016
(0, 0.6) 0.002918733 0.0001733 0.00005868
(0.3, 0.2) 0.002621451 0.0003985 0.00012684
(0.4, 0.6) 0.012447883 0.0012800 0.00000967
(0.7, 0.85) 0.020700332 0.0083391 0.00698802
(0.9, 0.9) 0.081198311 0.0232949 0.01644060
(x, y) $ |T_{5, 5, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}-f(x, y)| $ $ |T_{20, 20, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}-f(x, y)| $ $ |T_{30, 30, \lambda_{1}, \lambda_{2}}^{\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}-f(x, y)| $
(0, 0.1) 0.000347569 0.0000078 0.00000246
(0, 0.2) 0.000201932 0.0000009 0.00000039
(0, 0.4) 0.000862538 0.0000596 0.00002016
(0, 0.6) 0.002918733 0.0001733 0.00005868
(0.3, 0.2) 0.002621451 0.0003985 0.00012684
(0.4, 0.6) 0.012447883 0.0012800 0.00000967
(0.7, 0.85) 0.020700332 0.0083391 0.00698802
(0.9, 0.9) 0.081198311 0.0232949 0.01644060
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