doi: 10.3934/mfc.2022004
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Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials

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

*Corresponding author: Sompal Singh

Received  December 2021 Revised  January 2022 Early access February 2022

Karaisa [29] presented Jakimovski- Leviatan- Durrmeyer type operators by means of Appell polynomials. In a similar manner, Wani et al. [43] proposed a sequence of Jakimovski-Leviatan-Durrmeyer type operators involving Brenke type polynomials which include Appell polynomials and Hermite polynomials. We note that the definitions of the operators given in both these papers are not correct. In the present article, we introduce a Stancu variant of the operators considered in [43] after correcting their definition. The definition of the operator proposed in [29] may be similarly corrected. We establish the Korovkin type approximation theorem and the rate of convergence by means of the usual modulus of continuity, Peetre's K-functional and the class of Lipschitz type functions for our operators. Next, we discuss the Voronovskaja and Gr$ \ddot{u} $ss Voronovskaja type asymptotic theorems. Finally, we study the convergence of these operators in a weighted space and the Korovkin type weighted statistical approximation theorem.

Citation: Purshottam Narain Agrawal, Sompal Singh. Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022004
References:
[1]

T. Acar, Quantitative q-Voronovskaja and q-Grüss-Voronovskaja-type results for q-Szász operators, Georgian Math. J., 23 (2016), 459-468.  doi: 10.1515/gmj-2016-0007.

[2]

P. N. Agrawal, Ş. Y. G$\ddot{u}$ng$\ddot{o}$r and A. Kumar, Better degree of approximation by modified Bernstein-Durrmeyer type operators, Math. Found. Comp., (2021). doi: 10.3934/mfc.2021024.

[3]

R. Aktas, B. Çekim and F. Tasdelen, A Kantorovich-Stancu type generalization of Szász operators including Brenke type polynomials, J. Funct. Spaces Appl., (2013), Article ID 935430, 9 pp. doi: 10.1155/2013/935430.

[4]

F. Altomare and M. Campiti, Korovkin Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter, Berlin, New York, 1994. doi: 10.1515/9783110884586.

[5]

Ç. Atakut and I. Büyükyazici, Approximation by Kantorovich-Szász type operators based on Brenke type polynomials, Numer. Funct. Anal. Optim., 37 (2016), 1488-1502.  doi: 10.1080/01630563.2016.1216447.

[6]

C. BardaroP. L. ButzerR. L. Stens and G. Vinti, Approximation of the Whittaker sampling series in terms of an average modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl., 316 (2006), 269-306.  doi: 10.1016/j.jmaa.2005.04.042.

[7]

S. N. Bernstein, Dmonstration du théorème de Weierstrass fonde sur la calcul des probabilits, Commun. Soc. Math. Charkow Sr. 2 t., 13 (1912), 1-2. 

[8]

N. L. Braha, T. Mansour and M. Mursaleen, Some properties of Kantorovich-Stancu-type generalization of Szász operators including Brenke-type polynomials via power series summability method, J. Funct. Spaces., (2020), Art. ID 3480607, 15 pp. doi: 10.1155/2020/3480607.

[9]

P. L. Butzer and H. Karsli, Voronovskaya-type theorems for derivatives of the Bernstein-Chlodowsky polynomials and the Szász-Mirakyan operator, Comment. Math., 49 (2009), 33-58. 

[10]

P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, Springer Texts Electrical Eng., Springer, New York, (1993), 157–183.

[11]

P. Cardaliaguet and G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Netw, 5 (1992), 207-220.  doi: 10.1016/S0893-6080(05)80020-6.

[12]

F. Cucker and D. X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge University Press, 2007. doi: 10.1017/CBO9780511618796.

[13]

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

[14]

E. Erkuş-Duman and O. Duman, Statistical approximation properties of high order operators constructed with the Chan-Chyan-Srivastava polynomials, Appl. Math. Comput., 218 (2011), 1927-1933.  doi: 10.1016/j.amc.2011.07.004.

[15]

O. Duman and C. Orhan, Statistical approximation by positive linear operators, Stud. Math., 161 (2004), 187-197.  doi: 10.4064/sm161-2-6.

[16]

H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.  doi: 10.4064/cm-2-3-4-241-244.

[17]

A. D. Gadjiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P. P. Korovkin. Sov., Math. Dokl., 15 (1974), 1433-1436. 

[18]

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

[19]

A. D. Gadziev, On P. P. Korovkin type theorems, Mat. Zametki, 20 (1976), 781–786, (Transl. Math. Notes 56,995–998, (1978)). doi: 10.1007/BF01146928.

[20]

S. G. Gal and H. Gonska, Grüss and Grüss-Voronovskaya-type estimates for some Bernstein -type polynomials of reals and complex variables, Jean J. Approx., 7 (2015), 97-122. 

[21]

M. GoyalV. Gupta and P. N. Agrawal, Quantitative convergence results for a family of hybrid operators, Appl. Math. Comput., 271 (2015), 893-904.  doi: 10.1016/j.amc.2015.08.122.

[22]

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

[23]

V. Gupta, Error estimation for mixed summation-integral type operators, J. Math. Anal. Appl., 313 (2006), 632-641.  doi: 10.1016/j.jmaa.2005.05.017.

[24]

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

[25]

M. Ismail, On a generalization of Szász operators, Mathematica, 16 (1974), 259-267. 

[26]

A. Jakimovski and D. Leviatan, Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 11 (1969), 97-103. 

[27]

A. Kajla, Direct estimates of certain Mihesan-Durrmeyer type operators, Adv. Oper. Theory, 2 (2017), 162-178.  doi: 10.22034/aot.1612-1079.

[28]

A. Kajla and P. N. Agrawal, Approximation properties of Sz$\grave{a}$sz type operators based on Charlier polynomials, Turk. J. Math., 39 (2015), 990-1003.  doi: 10.3906/mat-1502-80.

[29]

A. Karaisa, Approximation by Durrmeyer type Jakimovski-Leviatan operators, Math. Methods Appl. Sci., 39 (2016), 2401-2410.  doi: 10.1002/mma.3650.

[30]

V. Karakaya and T. A. Chishti, Weighted statistical convergence, Iran. J. Sci. Technol., Trans. A Sci., 33 (2009), 219-223. 

[31]

H. Karsli, On multidimensional Urysohn type generalized sampling operators, Math. Found. Comp., 4 (2021), 271-280.  doi: 10.3934/mfc.2021015.

[32]

S. A. Mohiuddine, Statistical weighted A-summability with application to Korovkin's type approximation theorem, J. Inequal. Appl., 2016 (2016), Paper No. 101, 13 pp. doi: 10.1186/s13660-016-1040-1.

[33]

M. Mursaleen and K. J. Ansari, On Chlodowsky variant of Szász operators by Brenke type polynomials, Appl. Math. Comput., 271 (2015), 991-1003.  doi: 10.1016/j.amc.2015.08.123.

[34]

M. MursaleenV. KarkayaM. Ertürk and F. Gürsoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput., 218 (2012), 9132-9137.  doi: 10.1016/j.amc.2012.02.068.

[35]

M. Nasiruzzaman, N. Rao, S. Wazir and R. Kumar, Approximation on parametric extension of Baskakov-Durrmeyer operators on weighted space, J. Inequal. Appl., 2019 (2019), 103, 11 pp. doi: 10.1186/s13660-019-2055-1.

[36]

M. Ozhavzali and A. Olgun, On a modifed Szász-Mirakjan-Kantorovich operators in polynomial weighted spaces, Adv. Appl. Math. Sci., 13 (2014), 205-218. 

[37]

H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73-74. 

[38]

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

[39]

F. Taşdelen, R. Aktaş and A. Altin, A Kantorovich type of Szász operators including Brenke-type polynomials, Abstr. Appl. Anal., 2012 (2012), Article ID 867203. doi: 10.1155/2012/867203.

[40]

G. Ulusoy and T. Acar, q-Voronovskaya type theorems for q-Baskakov operators, Math. Methods Appl. Sci., 39 (2016), 3391-3401.  doi: 10.1002/mma.3784.

[41]

S. VarmaS. Sucu and G. İçóz, Generalization of Szász operators involving Brenke type polynomials, Comput. Math. Appl., 64 (2012), 121-127.  doi: 10.1016/j.camwa.2012.01.025.

[42]

S. Varma and F. Tasdelen, Szász type operators involving Charlier polynomials, Math. Comput. Model., 56 (2012), 118-122.  doi: 10.1016/j.mcm.2011.12.017.

[43]

S. A. Wani, M. Mursaleen and K. S. Nisar, Certain approximation properties of Brenke polynomials using Jakimovski-Leviatan operators, J. Inequal. Appl., 2021, Paper No. 104, 16 pp. doi: 10.1186/s13660-021-02639-2.

[44]

D.-X. Zhou, Deep distributed convolutional neural networks: Universality, Anal. Appl., 16 (2018), 895-919.  doi: 10.1142/S0219530518500124.

[45] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 1968. 

show all references

References:
[1]

T. Acar, Quantitative q-Voronovskaja and q-Grüss-Voronovskaja-type results for q-Szász operators, Georgian Math. J., 23 (2016), 459-468.  doi: 10.1515/gmj-2016-0007.

[2]

P. N. Agrawal, Ş. Y. G$\ddot{u}$ng$\ddot{o}$r and A. Kumar, Better degree of approximation by modified Bernstein-Durrmeyer type operators, Math. Found. Comp., (2021). doi: 10.3934/mfc.2021024.

[3]

R. Aktas, B. Çekim and F. Tasdelen, A Kantorovich-Stancu type generalization of Szász operators including Brenke type polynomials, J. Funct. Spaces Appl., (2013), Article ID 935430, 9 pp. doi: 10.1155/2013/935430.

[4]

F. Altomare and M. Campiti, Korovkin Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter, Berlin, New York, 1994. doi: 10.1515/9783110884586.

[5]

Ç. Atakut and I. Büyükyazici, Approximation by Kantorovich-Szász type operators based on Brenke type polynomials, Numer. Funct. Anal. Optim., 37 (2016), 1488-1502.  doi: 10.1080/01630563.2016.1216447.

[6]

C. BardaroP. L. ButzerR. L. Stens and G. Vinti, Approximation of the Whittaker sampling series in terms of an average modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl., 316 (2006), 269-306.  doi: 10.1016/j.jmaa.2005.04.042.

[7]

S. N. Bernstein, Dmonstration du théorème de Weierstrass fonde sur la calcul des probabilits, Commun. Soc. Math. Charkow Sr. 2 t., 13 (1912), 1-2. 

[8]

N. L. Braha, T. Mansour and M. Mursaleen, Some properties of Kantorovich-Stancu-type generalization of Szász operators including Brenke-type polynomials via power series summability method, J. Funct. Spaces., (2020), Art. ID 3480607, 15 pp. doi: 10.1155/2020/3480607.

[9]

P. L. Butzer and H. Karsli, Voronovskaya-type theorems for derivatives of the Bernstein-Chlodowsky polynomials and the Szász-Mirakyan operator, Comment. Math., 49 (2009), 33-58. 

[10]

P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, Springer Texts Electrical Eng., Springer, New York, (1993), 157–183.

[11]

P. Cardaliaguet and G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Netw, 5 (1992), 207-220.  doi: 10.1016/S0893-6080(05)80020-6.

[12]

F. Cucker and D. X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge University Press, 2007. doi: 10.1017/CBO9780511618796.

[13]

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

[14]

E. Erkuş-Duman and O. Duman, Statistical approximation properties of high order operators constructed with the Chan-Chyan-Srivastava polynomials, Appl. Math. Comput., 218 (2011), 1927-1933.  doi: 10.1016/j.amc.2011.07.004.

[15]

O. Duman and C. Orhan, Statistical approximation by positive linear operators, Stud. Math., 161 (2004), 187-197.  doi: 10.4064/sm161-2-6.

[16]

H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.  doi: 10.4064/cm-2-3-4-241-244.

[17]

A. D. Gadjiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P. P. Korovkin. Sov., Math. Dokl., 15 (1974), 1433-1436. 

[18]

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

[19]

A. D. Gadziev, On P. P. Korovkin type theorems, Mat. Zametki, 20 (1976), 781–786, (Transl. Math. Notes 56,995–998, (1978)). doi: 10.1007/BF01146928.

[20]

S. G. Gal and H. Gonska, Grüss and Grüss-Voronovskaya-type estimates for some Bernstein -type polynomials of reals and complex variables, Jean J. Approx., 7 (2015), 97-122. 

[21]

M. GoyalV. Gupta and P. N. Agrawal, Quantitative convergence results for a family of hybrid operators, Appl. Math. Comput., 271 (2015), 893-904.  doi: 10.1016/j.amc.2015.08.122.

[22]

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

[23]

V. Gupta, Error estimation for mixed summation-integral type operators, J. Math. Anal. Appl., 313 (2006), 632-641.  doi: 10.1016/j.jmaa.2005.05.017.

[24]

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

[25]

M. Ismail, On a generalization of Szász operators, Mathematica, 16 (1974), 259-267. 

[26]

A. Jakimovski and D. Leviatan, Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 11 (1969), 97-103. 

[27]

A. Kajla, Direct estimates of certain Mihesan-Durrmeyer type operators, Adv. Oper. Theory, 2 (2017), 162-178.  doi: 10.22034/aot.1612-1079.

[28]

A. Kajla and P. N. Agrawal, Approximation properties of Sz$\grave{a}$sz type operators based on Charlier polynomials, Turk. J. Math., 39 (2015), 990-1003.  doi: 10.3906/mat-1502-80.

[29]

A. Karaisa, Approximation by Durrmeyer type Jakimovski-Leviatan operators, Math. Methods Appl. Sci., 39 (2016), 2401-2410.  doi: 10.1002/mma.3650.

[30]

V. Karakaya and T. A. Chishti, Weighted statistical convergence, Iran. J. Sci. Technol., Trans. A Sci., 33 (2009), 219-223. 

[31]

H. Karsli, On multidimensional Urysohn type generalized sampling operators, Math. Found. Comp., 4 (2021), 271-280.  doi: 10.3934/mfc.2021015.

[32]

S. A. Mohiuddine, Statistical weighted A-summability with application to Korovkin's type approximation theorem, J. Inequal. Appl., 2016 (2016), Paper No. 101, 13 pp. doi: 10.1186/s13660-016-1040-1.

[33]

M. Mursaleen and K. J. Ansari, On Chlodowsky variant of Szász operators by Brenke type polynomials, Appl. Math. Comput., 271 (2015), 991-1003.  doi: 10.1016/j.amc.2015.08.123.

[34]

M. MursaleenV. KarkayaM. Ertürk and F. Gürsoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput., 218 (2012), 9132-9137.  doi: 10.1016/j.amc.2012.02.068.

[35]

M. Nasiruzzaman, N. Rao, S. Wazir and R. Kumar, Approximation on parametric extension of Baskakov-Durrmeyer operators on weighted space, J. Inequal. Appl., 2019 (2019), 103, 11 pp. doi: 10.1186/s13660-019-2055-1.

[36]

M. Ozhavzali and A. Olgun, On a modifed Szász-Mirakjan-Kantorovich operators in polynomial weighted spaces, Adv. Appl. Math. Sci., 13 (2014), 205-218. 

[37]

H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73-74. 

[38]

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

[39]

F. Taşdelen, R. Aktaş and A. Altin, A Kantorovich type of Szász operators including Brenke-type polynomials, Abstr. Appl. Anal., 2012 (2012), Article ID 867203. doi: 10.1155/2012/867203.

[40]

G. Ulusoy and T. Acar, q-Voronovskaya type theorems for q-Baskakov operators, Math. Methods Appl. Sci., 39 (2016), 3391-3401.  doi: 10.1002/mma.3784.

[41]

S. VarmaS. Sucu and G. İçóz, Generalization of Szász operators involving Brenke type polynomials, Comput. Math. Appl., 64 (2012), 121-127.  doi: 10.1016/j.camwa.2012.01.025.

[42]

S. Varma and F. Tasdelen, Szász type operators involving Charlier polynomials, Math. Comput. Model., 56 (2012), 118-122.  doi: 10.1016/j.mcm.2011.12.017.

[43]

S. A. Wani, M. Mursaleen and K. S. Nisar, Certain approximation properties of Brenke polynomials using Jakimovski-Leviatan operators, J. Inequal. Appl., 2021, Paper No. 104, 16 pp. doi: 10.1186/s13660-021-02639-2.

[44]

D.-X. Zhou, Deep distributed convolutional neural networks: Universality, Anal. Appl., 16 (2018), 895-919.  doi: 10.1142/S0219530518500124.

[45] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 1968. 
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