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doi: 10.3934/mfc.2021016
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Convergence of derivative of Szász type operators involving Charlier polynomials

Purshottam N. Agrawal 1, , Thakur Ashok K. Sinha 2,3, and  Avinash Sharma 2,3,

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 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.

Keywords: Szász operators, Charlier polynomials, rate of convergence, simultaneous approximation, modulus of continuity, linear combination.
Mathematics Subject Classification: 41A36, 41A25, 41A28, 26A15.
Citation: Purshottam N. Agrawal, Thakur Ashok K. Sinha, Avinash Sharma. Convergence of derivative of Szász type operators involving Charlier polynomials. Mathematical Foundations of Computing, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. G. Lorentz, Bernstein polynomials, University of Toronto Press, Toronto, (1953), 130pp.  Google Scholar

[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.  Google Scholar

[14]

H. N. Mhaskar and D. V. Pai, Fundamentals of Approximation Theory, Narosa Publ. House, New Delhi, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[19]

A. F. Timan, Theory of Approximation of Functions of a Real Variable, Hindustan Pub. Co., New Delhi, 1966. Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. G. Lorentz, Bernstein polynomials, University of Toronto Press, Toronto, (1953), 130pp.  Google Scholar

[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.  Google Scholar

[14]

H. N. Mhaskar and D. V. Pai, Fundamentals of Approximation Theory, Narosa Publ. House, New Delhi, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[19]

A. F. Timan, Theory of Approximation of Functions of a Real Variable, Hindustan Pub. Co., New Delhi, 1966. Google Scholar

[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.  Google Scholar

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