• Previous Article
    Some generalizations of delay integral inequalities of Gronwall-Bellman type with power and their applications
  • MFC Home
  • This Issue
  • Next Article
    Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors
doi: 10.3934/mfc.2021016
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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, doi: 10.3934/mfc.2021016
References:
[1]

P. N. AgrawalB. 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. ChauhanB. 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. DeoN. 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. GuptaN. 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. SinhaP. 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. AgrawalB. 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. ChauhanB. 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. DeoN. 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. GuptaN. 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. SinhaP. 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

[1]

Mohd Qasim, Mohd Shanawaz Mansoori, Asif Khan, Zaheer Abbas, Mohammad Mursaleen. Convergence of modified Szász-Mirakyan-Durrmeyer operators depending on certain parameters. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021027

[2]

Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129

[3]

Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1

[4]

Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465

[5]

Benjamin Jourdain, Julien Reygner. Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4963-4996. doi: 10.3934/dcds.2016015

[6]

Thomas Gauthier, Gabriel Vigny. Distribution of postcritically finite polynomials Ⅱ: Speed of convergence. Journal of Modern Dynamics, 2017, 11: 57-98. doi: 10.3934/jmd.2017004

[7]

Danilo Costarelli. Preface: Special issue on approximation by linear and nonlinear operators with applications. Part I. Mathematical Foundations of Computing, 2021, 4 (4) : i-ii. doi: 10.3934/mfc.2021028

[8]

Domingo Tarzia, Carolina Bollo, Claudia Gariboldi. Convergence of simultaneous distributed-boundary parabolic optimal control problems. Evolution Equations & Control Theory, 2020, 9 (4) : 1187-1201. doi: 10.3934/eect.2020045

[9]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190

[10]

Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219

[11]

Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial & Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199

[12]

Shahad Al-azzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 227-245. doi: 10.3934/dcdsb.2017012

[13]

Armand Bernou. A semigroup approach to the convergence rate of a collisionless gas. Kinetic & Related Models, 2020, 13 (6) : 1071-1106. doi: 10.3934/krm.2020038

[14]

Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503

[15]

Nathan Albin, Nethali Fernando, Pietro Poggi-Corradini. Modulus metrics on networks. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 1-17. doi: 10.3934/dcdsb.2018161

[16]

He Zhang, John Harlim, Xiantao Li. Estimating linear response statistics using orthogonal polynomials: An RKHS formulation. Foundations of Data Science, 2020, 2 (4) : 443-485. doi: 10.3934/fods.2020021

[17]

Ling-Xiong Han, Wen-Hui Li, Feng Qi. Approximation by multivariate Baskakov–Kantorovich operators in Orlicz spaces. Electronic Research Archive, 2020, 28 (2) : 721-738. doi: 10.3934/era.2020037

[18]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753

[19]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851

[20]

Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008

 Impact Factor: 

Metrics

  • PDF downloads (69)
  • HTML views (59)
  • Cited by (0)

[Back to Top]