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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 |
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 [
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. |
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. |
[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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