November  2022, 5(4): 315-330. doi: 10.3934/mfc.2022007

Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour

1. 

Department of Mathematics, Faculty of Science, Indira Gandhi National Tribal University, Amarkantak, Madhya Pradesh, 484887 India

2. 

Department of Mathematics Shree Guru Gobind Singh, Tricentenary University, Gurugram-122505 Haryana India

3. 

Department of Mathematics, Hansraj College, Mathma Hans Raj Marg, Malka Ganj, New Delhi, Delhi 110007, India

*Corresponding author: Vishnu Narayan Mishra and Nadeem Rao

Received  October 2021 Revised  February 2022 Published  November 2022 Early access  March 2022

The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz$ \acute{a} $sz-Schurer-Beta type operators to approximate a class of Lebesgue integrable functions. Moreover, we calculate basic estimates and central moments for these sequences of operators. Further, rapidity of convergence and order of approximation are investigated in terms of Korovkin theorem and modulus of smoothess. In subsequent section, local and global approximation properties are studied in various functional spaces.

Citation: Mohd Raiz, Amit Kumar, Vishnu Narayan Mishra, Nadeem Rao. Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour. Mathematical Foundations of Computing, 2022, 5 (4) : 315-330. doi: 10.3934/mfc.2022007
References:
[1]

T. AcarS. A. Mohiuddine and M. Mursaleen, Approximation by (p, q)-Baskakov-Durrmeyer-Stancu operators, Complex Anal. Oper. Theory, 12 (2018), 1453-1468.  doi: 10.1007/s11785-016-0633-5.

[2]

A. M. Acu, Stancu Schurer Kantorovich operators based on q-integers, Appl. Math. Comput., 259 (2015), 896-907.  doi: 10.1016/j.amc.2015.03.032.

[3]

A. M. Acu and I. Rasa, Estimates for the differences of positive linear operators and their derivatives, Num. Alg, 85 (2020), 191-208.  doi: 10.1007/s11075-019-00809-4.

[4]

P. N. Agrawal and A. Kajla, Szász-Durrmeyer type operators based on Charlier polynomials, Appl. Math. Comput, 268 (2015), 1001-1014.  doi: 10.1016/j.amc.2015.06.126.

[5]

P. N. AgrwalT. A. Sinha and A. Sharma, Convergence of derivative of Szasz type operators involving Charlier polynomials, Math. Fund. Computin, 5 (2022), 1-15. 

[6]

A. Alotaibi, M. Nasiruzzaman and M. Mursaleen, A Dunkl type generalization of Szász operators via post-quantum calculus, J. Inequal. Appl., 2018 (2018), Paper No. 287, 15 pp. doi: 10.1186/s13660-018-1878-5.

[7]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff. De Gruyter Studies in Mathematics, 17. Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.

[8]

S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités, Commun. Soc. Math. Kharkow, 2 (1912), 1-2. 

[9]

N. L. Braha, Some properties of Baskakov-Schurer-Szász operators via power summability methods, Quaes. Math., 42 (2019), 1411-1426.  doi: 10.2989/16073606.2018.1523248.

[10]

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der mathematischen Wissenschaften, 303. Springer-Verlag, Berlin, 1993.

[11]

O. Duman and C. Orhan, Statistical approximation by positive linear operators, Stud. Math., 16 (2004), 187-197.  doi: 10.4064/sm161-2-6.

[12]

A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (2002), 129-138.  doi: 10.1216/rmjm/1030539612.

[13]

A. D. Gadziev, Theorems of the type of P.P. Korovkin's theorems, Mat. Zame., 20 (1976), 781-786. 

[14]

X. GuoL. X. Li and Q. Wu, Modeling interactive components by coordinate Kernel polynomial models, Math. Fund. Computing, 3 (2020), 263-277.  doi: 10.3934/mfc.2020010.

[15]

Z. C. GuoD. H. XiangX. Guo and D. X. Zhou, Threshold spectral algorithms for sparse approximations, Anal. Appl., 15 (2017), 433-455.  doi: 10.1142/S0219530517500026.

[16]

E. Ibikli and E. A. Gadjieva, The order of approximation of some unbounded functions by the sequence of positive linear operators, Turk. J. Math., 19 (1995), 331-337. 

[17]

U. Kadak and S. A. Mohiuddine, Generalized statistically almost convergence based on the difference operator which includes the (p, q)-gamma function and related approximation theorems Results in Math, Results in Maths, 73 (2018), Paper No. 9, 31 pp. doi: 10.1007/s00025-018-0789-6.

[18]

A. Kajla and P. N. Agrawal, Approximation properties of Szász type operators based on Charlier polynomials, Turk. J. Math., 39 (2015), 990-1003.  doi: 10.3906/mat-1502-80.

[19]

A. Kajla and P. N. Agrawal, Szász-Kantorovich type operatorsbased on Charlier polynomials, Kyungpook Math. J., 56 (2016), 877-897.  doi: 10.5666/KMJ.2016.56.3.877.

[20]

B. Lenze, On Lipschitz type maximal functions and their smoothness spaces, Nederl Akad. Indag. Math., 50 (1988), 53-63. 

[21]

V. N. Mishra and R. B. Gandhi, A Summation-Integral type modification of Szász - Mirakjan operators, Math. Methods Appl. Sci., 40 (2017), 175-182.  doi: 10.1002/mma.3977.

[22]

V.N. Mishra, K. Khatri, L.N. Mishra, Deepmala, Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, Jnrl of Ineq. and Appli., 2013.

[23]

V.N. MishraK. Khatri and L.N. Mishra, Statistical approximation by Kantorovich type Discrete $q-$Beta operators,, Advn. in Diff. Eqn., 345 (2013), 2013-345. 

[24]

R.B. GandhiDe epmala and V.N. Mishra, Local and global results for modified Szász - Mirakjan operators, Math., Method. Appl. Sci., 40 (2017), 2491-2504. 

[25]

V.N. Mishra, A.R. Devdhara, R.B. Gandhi, Global Approximation Theorems for the Generalized Sz$\acute{a}$sz-Mirakjan type Operators in Exponential Weight Spaces, Appl. Math. Comp., 336, (2018), 206–214.

[26]

S. A. MohiuddineT. Acar and A. Alotaibi, Construction of a new family of Bernstein Kantorovich operators, Math. Methods Appl. Sci., 40 (2017), 7749-7759.  doi: 10.1002/mma.4559.

[27]

M. MursaleenK. J. Ansari and A. Khan, Approximation by a Kantorovich type q -Bernstein-Stancu operators, Complex Anal. Oper. Theory, 11 (2017), 85-107.  doi: 10.1007/s11785-016-0572-1.

[28]

M. MursaleenA. Naaz and A. Khan, Approximation and error estimations by King type (p, q)-Szász-Mirakjan Kantorovich operators, Appl. Math. Comput., 348 (2019), 175-185.  doi: 10.1016/j.amc.2018.11.044.

[29]

N. RaoM. HeshamuddinM. Shadab and A. Srivastava, Approximation properties of bivariate Szász Durrmeyer operators via Dunkl analogu, Filomat, 35 (2021), 4515-4532.  doi: 10.2298/FIL2113515R.

[30]

N. Rao and M. Nasiruzaman, A generalized Dunkl type modification of Phillips operators, J. Inequal. Appl., 2018 (2018), Paper No. 323, 12 pp. doi: 10.1186/s13660-018-1909-2.

[31]

M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Oper. Theory Adv. Appl., 73 (1994), 369-396. 

[32]

F. Schurer, Linear Positive Operators in Approximation Theory, Dissertation, Technological University of Delft, 1965 Uitgeverij Waltman, Delft 1965, 79 pp.

[33]

O. Shisha and B. Bond, The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci., 60 (1968), 1196-1200.  doi: 10.1073/pnas.60.4.1196.

[34]

M.A. $\ddot{o}$zarslan and H. Aktu$\breve{g}$lu, Local approximation for certain King type operators, Filomat 27., (2013), 173-181. 

[35]

S. Sucu, Dunkl analogue of Szász operators, Appl. Math. Comput., 244 (2014), 42-48.  doi: 10.1016/j.amc.2014.06.088.

[36]

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

[37]

A. Wafi and N. Rao, A generalization of Szász-type operators which preserves constant and quadratic test functions, Cogent. Math., 3 (2016), 111-118.  doi: 10.1080/23311835.2016.1227023.

[38]

A. Wafi and N. Rao, Szász-Durrmeyer operators based on Dunkl analogue, Complex Anal. Oper. Theory, 12 (2018), 1519-1536.  doi: 10.1007/s11785-017-0647-7.

[39]

A. Wafi and N. Rao, On Kantorovich form of generalized Szász-type operators using Charlier polynomia, Korean J. Math., 25 (2017), 99-116.  doi: 10.11568/kjm.2017.25.1.99.

show all references

References:
[1]

T. AcarS. A. Mohiuddine and M. Mursaleen, Approximation by (p, q)-Baskakov-Durrmeyer-Stancu operators, Complex Anal. Oper. Theory, 12 (2018), 1453-1468.  doi: 10.1007/s11785-016-0633-5.

[2]

A. M. Acu, Stancu Schurer Kantorovich operators based on q-integers, Appl. Math. Comput., 259 (2015), 896-907.  doi: 10.1016/j.amc.2015.03.032.

[3]

A. M. Acu and I. Rasa, Estimates for the differences of positive linear operators and their derivatives, Num. Alg, 85 (2020), 191-208.  doi: 10.1007/s11075-019-00809-4.

[4]

P. N. Agrawal and A. Kajla, Szász-Durrmeyer type operators based on Charlier polynomials, Appl. Math. Comput, 268 (2015), 1001-1014.  doi: 10.1016/j.amc.2015.06.126.

[5]

P. N. AgrwalT. A. Sinha and A. Sharma, Convergence of derivative of Szasz type operators involving Charlier polynomials, Math. Fund. Computin, 5 (2022), 1-15. 

[6]

A. Alotaibi, M. Nasiruzzaman and M. Mursaleen, A Dunkl type generalization of Szász operators via post-quantum calculus, J. Inequal. Appl., 2018 (2018), Paper No. 287, 15 pp. doi: 10.1186/s13660-018-1878-5.

[7]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff. De Gruyter Studies in Mathematics, 17. Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.

[8]

S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités, Commun. Soc. Math. Kharkow, 2 (1912), 1-2. 

[9]

N. L. Braha, Some properties of Baskakov-Schurer-Szász operators via power summability methods, Quaes. Math., 42 (2019), 1411-1426.  doi: 10.2989/16073606.2018.1523248.

[10]

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der mathematischen Wissenschaften, 303. Springer-Verlag, Berlin, 1993.

[11]

O. Duman and C. Orhan, Statistical approximation by positive linear operators, Stud. Math., 16 (2004), 187-197.  doi: 10.4064/sm161-2-6.

[12]

A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (2002), 129-138.  doi: 10.1216/rmjm/1030539612.

[13]

A. D. Gadziev, Theorems of the type of P.P. Korovkin's theorems, Mat. Zame., 20 (1976), 781-786. 

[14]

X. GuoL. X. Li and Q. Wu, Modeling interactive components by coordinate Kernel polynomial models, Math. Fund. Computing, 3 (2020), 263-277.  doi: 10.3934/mfc.2020010.

[15]

Z. C. GuoD. H. XiangX. Guo and D. X. Zhou, Threshold spectral algorithms for sparse approximations, Anal. Appl., 15 (2017), 433-455.  doi: 10.1142/S0219530517500026.

[16]

E. Ibikli and E. A. Gadjieva, The order of approximation of some unbounded functions by the sequence of positive linear operators, Turk. J. Math., 19 (1995), 331-337. 

[17]

U. Kadak and S. A. Mohiuddine, Generalized statistically almost convergence based on the difference operator which includes the (p, q)-gamma function and related approximation theorems Results in Math, Results in Maths, 73 (2018), Paper No. 9, 31 pp. doi: 10.1007/s00025-018-0789-6.

[18]

A. Kajla and P. N. Agrawal, Approximation properties of Szász type operators based on Charlier polynomials, Turk. J. Math., 39 (2015), 990-1003.  doi: 10.3906/mat-1502-80.

[19]

A. Kajla and P. N. Agrawal, Szász-Kantorovich type operatorsbased on Charlier polynomials, Kyungpook Math. J., 56 (2016), 877-897.  doi: 10.5666/KMJ.2016.56.3.877.

[20]

B. Lenze, On Lipschitz type maximal functions and their smoothness spaces, Nederl Akad. Indag. Math., 50 (1988), 53-63. 

[21]

V. N. Mishra and R. B. Gandhi, A Summation-Integral type modification of Szász - Mirakjan operators, Math. Methods Appl. Sci., 40 (2017), 175-182.  doi: 10.1002/mma.3977.

[22]

V.N. Mishra, K. Khatri, L.N. Mishra, Deepmala, Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, Jnrl of Ineq. and Appli., 2013.

[23]

V.N. MishraK. Khatri and L.N. Mishra, Statistical approximation by Kantorovich type Discrete $q-$Beta operators,, Advn. in Diff. Eqn., 345 (2013), 2013-345. 

[24]

R.B. GandhiDe epmala and V.N. Mishra, Local and global results for modified Szász - Mirakjan operators, Math., Method. Appl. Sci., 40 (2017), 2491-2504. 

[25]

V.N. Mishra, A.R. Devdhara, R.B. Gandhi, Global Approximation Theorems for the Generalized Sz$\acute{a}$sz-Mirakjan type Operators in Exponential Weight Spaces, Appl. Math. Comp., 336, (2018), 206–214.

[26]

S. A. MohiuddineT. Acar and A. Alotaibi, Construction of a new family of Bernstein Kantorovich operators, Math. Methods Appl. Sci., 40 (2017), 7749-7759.  doi: 10.1002/mma.4559.

[27]

M. MursaleenK. J. Ansari and A. Khan, Approximation by a Kantorovich type q -Bernstein-Stancu operators, Complex Anal. Oper. Theory, 11 (2017), 85-107.  doi: 10.1007/s11785-016-0572-1.

[28]

M. MursaleenA. Naaz and A. Khan, Approximation and error estimations by King type (p, q)-Szász-Mirakjan Kantorovich operators, Appl. Math. Comput., 348 (2019), 175-185.  doi: 10.1016/j.amc.2018.11.044.

[29]

N. RaoM. HeshamuddinM. Shadab and A. Srivastava, Approximation properties of bivariate Szász Durrmeyer operators via Dunkl analogu, Filomat, 35 (2021), 4515-4532.  doi: 10.2298/FIL2113515R.

[30]

N. Rao and M. Nasiruzaman, A generalized Dunkl type modification of Phillips operators, J. Inequal. Appl., 2018 (2018), Paper No. 323, 12 pp. doi: 10.1186/s13660-018-1909-2.

[31]

M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Oper. Theory Adv. Appl., 73 (1994), 369-396. 

[32]

F. Schurer, Linear Positive Operators in Approximation Theory, Dissertation, Technological University of Delft, 1965 Uitgeverij Waltman, Delft 1965, 79 pp.

[33]

O. Shisha and B. Bond, The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci., 60 (1968), 1196-1200.  doi: 10.1073/pnas.60.4.1196.

[34]

M.A. $\ddot{o}$zarslan and H. Aktu$\breve{g}$lu, Local approximation for certain King type operators, Filomat 27., (2013), 173-181. 

[35]

S. Sucu, Dunkl analogue of Szász operators, Appl. Math. Comput., 244 (2014), 42-48.  doi: 10.1016/j.amc.2014.06.088.

[36]

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

[37]

A. Wafi and N. Rao, A generalization of Szász-type operators which preserves constant and quadratic test functions, Cogent. Math., 3 (2016), 111-118.  doi: 10.1080/23311835.2016.1227023.

[38]

A. Wafi and N. Rao, Szász-Durrmeyer operators based on Dunkl analogue, Complex Anal. Oper. Theory, 12 (2018), 1519-1536.  doi: 10.1007/s11785-017-0647-7.

[39]

A. Wafi and N. Rao, On Kantorovich form of generalized Szász-type operators using Charlier polynomia, Korean J. Math., 25 (2017), 99-116.  doi: 10.11568/kjm.2017.25.1.99.

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