# American Institute of Mathematical Sciences

February  2022, 5(1): 1-15. doi: 10.3934/mfc.2021016

## Convergence of derivative of Szász type operators involving Charlier polynomials

 1 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India 2 Post Graduate Department of Mathematics, Patliputra University, Patna 800020, India 3 Post Graduate Department of Mathematics, Magadh University, Bodh Gaya, Gaya 824234, India

Received  May 2021 Revised  July 2021 Published  February 2022 Early access  September 2021

The paper deals with the approximation of first order derivative of a function by the first order derivative of Szász-type operators based on Charlier polynomials introduced by Varma and Taşdelen [20]. The uniform convergence theorem, Voronovskaja type asymptotic theorem and an estimate of error in terms of the second order modulus of continuity of the derivative of the function are investigated. Further, it is shown that linear combinations of the derivative of the above operators converge to the derivative of function at a faster rate. Finally, an estimate of error in the approximation is obtained in terms of the $(2k+2)th$ order modulus of continuity using Steklov mean.

Citation: Purshottam N. Agrawal, Thakur Ashok K. Sinha, Avinash Sharma. Convergence of derivative of Szász type operators involving Charlier polynomials. Mathematical Foundations of Computing, 2022, 5 (1) : 1-15. doi: 10.3934/mfc.2021016
##### References:
 [1] P. N. Agrawal, B. Baxhaku and R. Chauhan, The approximation of bivariate Chlodowsky-Szász- Kantorovich-Charlier type operators, J. Inequal. Appl., 2017 (2017), 195.  doi: 10.1186/s13660-017-1465-1. [2] R. Chauhan, B. Baxhaku and P. N. Agrawal, Szász type operators involving Charlier polynomials of blending type, Complex Anal. Oper. Theory, 13 (2019), 1197-1226.  doi: 10.1007/s11785-018-0854-x. [3] T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and Its Applications, Vol. 13. Gordon and Breach Science Publishers, New York-London-Paris, 1978. [4] N. Deo, N. Bharadwaj and S. P. Singh, Simultaneous approximation on generalized Bernstein-Durrmeyer operators, Afr. Mat., 24 (2013), 77-82.  doi: 10.1007/s13370-011-0041-y. [5] V. Gupta, N. Deo and X. M. Zeng, Simultaneous approximation for Szász-Mirakjan -Stancu-Durrmeyer operators, Anal. Theory Appl., 29 (2013), 86-96.  doi: 10.4208/ata.2013.v29.n1.9. [6] V. Gupta, M. A. Noor, M. S. Beniwal and M. K. Gupta, On simultaneous approximation for certain Baskakov-Durrmeyer type operators, JIPAM. J. Inequal. Pure Appl. Math., 7 (2006), 15pp. [7] M. Heilmann and M. W. Müller, On simultaneous approximation by the method of Baskakov- Durrmeyer operators, Numer. Funct. Anal. Optim., 10 (1989), 127-138.  doi: 10.1080/01630568908816295. [8] M. Heilmann and M. W. Müller, Direct and converse results on simultaneous approximation by the method of Bernstein-Durrmeyer operators, Algorithms for Approximation, II(Shrivenham, 1988), Chapman and Hall, London, (1990), 107–116. [9] 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. [10] A. Kajla and P. N. Agrawal, Szász-Durrmeyer type operators based on Charlier polynomials, Appl. Math. Comput., 268 (2015), 1001-1014.  doi: 10.1016/j.amc.2015.06.126. [11] A. Kajla and P. N. Agrawal, Szász-Kantorovitch type operators based on Charlier polynomials, Kyungpook Math. J., 56 (2016), 877-897.  doi: 10.5666/KMJ.2016.56.3.877. [12] G. G. Lorentz, Bernstein polynomials, University of Toronto Press, Toronto, (1953), 130pp. [13] C. P. May, Saturation and inverse theorems for combinations of a class of exponential type operators, Canad. J. Math., 28 (1976), 1224-1250.  doi: 10.4153/CJM-1976-123-8. [14] H. N. Mhaskar and D. V. Pai, Fundamentals of Approximation Theory, Narosa Publ. House, New Delhi, 2000. [15] R. K. S. Rathore, (L, p)-summability of a multiply differentiated Fourier series, Nederl. Akad. Wetensch. Proc. Ser. A, Indag. Math., 38 (1976), 217–230. [16] R. K. S. Rathore, Lipschitz-Nikolskii constants and asymptotic simultaneous approximation of the $M_n$-operators, Aequationes Math., 18 (1978), 206-217.  doi: 10.1007/BF01844075. [17] R. K. S. Rathore and O. P. Singh, On convergence of derivatives of Post-Widder operators, Indian J. Pure Appl. Math., 11 (1980), 547-561. [18] R. P. Sinha, P. N. Agrawal and V. Gupta, On simultaneous approximation by modified Baskakov operators, Bull. Soc. Math. Belg. Ser. B, 43 (1991), 217-231. [19] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Hindustan Pub. Co., New Delhi, 1966. [20] S. Varma and F. Taşdelen, Szász type operators involving Charlier polynomials, Math. Comput. Modelling, 56 (2012), 118-122.  doi: 10.1016/j.mcm.2011.12.017.

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##### References:
 [1] P. N. Agrawal, B. Baxhaku and R. Chauhan, The approximation of bivariate Chlodowsky-Szász- Kantorovich-Charlier type operators, J. Inequal. Appl., 2017 (2017), 195.  doi: 10.1186/s13660-017-1465-1. [2] R. Chauhan, B. Baxhaku and P. N. Agrawal, Szász type operators involving Charlier polynomials of blending type, Complex Anal. Oper. Theory, 13 (2019), 1197-1226.  doi: 10.1007/s11785-018-0854-x. [3] T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and Its Applications, Vol. 13. Gordon and Breach Science Publishers, New York-London-Paris, 1978. [4] N. Deo, N. Bharadwaj and S. P. Singh, Simultaneous approximation on generalized Bernstein-Durrmeyer operators, Afr. Mat., 24 (2013), 77-82.  doi: 10.1007/s13370-011-0041-y. [5] V. Gupta, N. Deo and X. M. Zeng, Simultaneous approximation for Szász-Mirakjan -Stancu-Durrmeyer operators, Anal. Theory Appl., 29 (2013), 86-96.  doi: 10.4208/ata.2013.v29.n1.9. [6] V. Gupta, M. A. Noor, M. S. Beniwal and M. K. Gupta, On simultaneous approximation for certain Baskakov-Durrmeyer type operators, JIPAM. J. Inequal. Pure Appl. Math., 7 (2006), 15pp. [7] M. Heilmann and M. W. Müller, On simultaneous approximation by the method of Baskakov- Durrmeyer operators, Numer. Funct. Anal. Optim., 10 (1989), 127-138.  doi: 10.1080/01630568908816295. [8] M. Heilmann and M. W. Müller, Direct and converse results on simultaneous approximation by the method of Bernstein-Durrmeyer operators, Algorithms for Approximation, II(Shrivenham, 1988), Chapman and Hall, London, (1990), 107–116. [9] 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. [10] A. Kajla and P. N. Agrawal, Szász-Durrmeyer type operators based on Charlier polynomials, Appl. Math. Comput., 268 (2015), 1001-1014.  doi: 10.1016/j.amc.2015.06.126. [11] A. Kajla and P. N. Agrawal, Szász-Kantorovitch type operators based on Charlier polynomials, Kyungpook Math. J., 56 (2016), 877-897.  doi: 10.5666/KMJ.2016.56.3.877. [12] G. G. Lorentz, Bernstein polynomials, University of Toronto Press, Toronto, (1953), 130pp. [13] C. P. May, Saturation and inverse theorems for combinations of a class of exponential type operators, Canad. J. Math., 28 (1976), 1224-1250.  doi: 10.4153/CJM-1976-123-8. [14] H. N. Mhaskar and D. V. Pai, Fundamentals of Approximation Theory, Narosa Publ. House, New Delhi, 2000. [15] R. K. S. Rathore, (L, p)-summability of a multiply differentiated Fourier series, Nederl. Akad. Wetensch. Proc. Ser. A, Indag. Math., 38 (1976), 217–230. [16] R. K. S. Rathore, Lipschitz-Nikolskii constants and asymptotic simultaneous approximation of the $M_n$-operators, Aequationes Math., 18 (1978), 206-217.  doi: 10.1007/BF01844075. [17] R. K. S. Rathore and O. P. Singh, On convergence of derivatives of Post-Widder operators, Indian J. Pure Appl. Math., 11 (1980), 547-561. [18] R. P. Sinha, P. N. Agrawal and V. Gupta, On simultaneous approximation by modified Baskakov operators, Bull. Soc. Math. Belg. Ser. B, 43 (1991), 217-231. [19] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Hindustan Pub. Co., New Delhi, 1966. [20] S. Varma and F. Taşdelen, Szász type operators involving Charlier polynomials, Math. Comput. Modelling, 56 (2012), 118-122.  doi: 10.1016/j.mcm.2011.12.017.
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