A summation-integral type modification of Szasz-Mirakjan-Stancu operators

Authors

  • Vishnu Narayan Mishra Sardar Vallabhbhai National Institute of Technology, Surat - 395007, Gujarat, India
  • Rajiv B. Gandhi Birla Vishvakarma Mahavidyalaya, Vallabh Vidyanagar - 388120, Gujarat, India
  • Ram N. Mohapatra University of Central Florida, Orlando, FL. 32816, USA

DOI:

https://doi.org/10.33993/jnaat451-1075

Keywords:

Szasz-Mirakjan operators, the Korovkin-type approximation result, K-functional, modulus of smoothness, Voronovskaja-type result
Abstract views: 353

Abstract

In this paper we introduce a summation-integral type modification of Szasz-Mirakjan-Stancu operators. Calculation of moments, density theorem, a direct result and a Voronovskaja-type result are obtained for the operators.

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Author Biographies

Vishnu Narayan Mishra, Sardar Vallabhbhai National Institute of Technology, Surat - 395007, Gujarat, India

Applied Mathematics & Humanities Department

Rajiv B. Gandhi, Birla Vishvakarma Mahavidyalaya, Vallabh Vidyanagar - 388120, Gujarat, India

Department of Mathematics

Ram N. Mohapatra, University of Central Florida, Orlando, FL. 32816, USA

Department of Mathematics

References

T. Acar, L.N. Mishra and V.N. Mishra, Simultaneous approximation for generalized Srivastava-Gupta operator, J. Function Spaces, Article ID 936308, 12 pages, 2015, http://dx.doi.org/10.1155/2015/936308 DOI: https://doi.org/10.1155/2015/936308

O. Agratini, On approximation process of integral type, Appl. Math. Comput., 236 (2014) pp. 195–201, http://dx.doi.org/10.1016/j.amc.2014.03.052 DOI: https://doi.org/10.1016/j.amc.2014.03.052

P.N. Agrawal and H.S. Kasana, On simultaneous approximation by Szasz-Mirakjan operators. Bull. Inst. Math. Acad. Sinica 22 (1994), pp. 181–188.

P. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications , Walter de Gruyter, Berlin - New York, 1994. DOI: https://doi.org/10.1515/9783110884586

M. Becker, Global approximation theorems for Szasz-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27 (1978), pp. 127–142, http://dx.doi.org/10.1512/iumj.1978.27.27011

M. Becker, D. Kucharski and R.J. Nessel, Global approximation theorems for Szasz-Mirakjan and Baskakov operators in exponential weight spaces, In: Linear Spaces and Approximation (Proc. Conf. Oberwolfach, 1977), Birkh ̃auser Verlag, Basel. DOI: https://doi.org/10.1007/978-3-0348-7180-8_28

P.L. Butzer, On the extension of Bernstein polynomials to the infinite interval. Proc. Amer. Math. Soc., 5 (1954), pp. 547–553, http://dx.doi.org/10.2307/2032032 DOI: https://doi.org/10.1090/S0002-9939-1954-0063483-7

M.M. Derriennic, Sur l’approximation des fonctions d’une ou plusieurs variables par des polynomes de Bernstein modifi ́es et application au probl ́eme des moments, These de 3e cycle, Universite de Rennes, 1978.

J.L. Durrmeyer, Une formule d’inversion de la transformee de Laplace: Applications a la theorie des moments , These de 3e cycle, Faculte des Sciences de l’Universite de Paris, 1967.

A.R. Gairola, Deepmala, L.N. Mishra, Rate of approximation by finite iterates of q-Durrmeyer operators, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 86 (2016) no. 2, pp. 229-234, http://dx.doi.org/10.1007/s40010-016-0267-z DOI: https://doi.org/10.1007/s40010-016-0267-z

I. Gurhan and R.N. Mohapatra, Approximation properties by q-Durrmeyer-Stancu operators, Anal. Theory Appl. 29 (4) (2013), pp. 373-383. DOI: https://doi.org/10.4208/ata.2013.v29.n4.6

T. Hermann, On the Szasz-Mirakjan operator, Acta Math. Sci. Acad. Hungar 32 (1-2) (1978), pp. 163-173, http://dx.doi.org/10.1007/BF01902211 DOI: https://doi.org/10.1007/BF01902211

H.M. Srivastava, Z. Finta and V. Gupta, Direct results for a certain family of summation-integral type operators, Appl. Math. Comput. 190 (2007), pp. 449-457, http://dx.doi.org/10.1016/j.amc.2007.01.039 DOI: https://doi.org/10.1016/j.amc.2007.01.039

H.M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling 37 (2003), pp. 1307-1315, http://dx.doi.org/10.1016/S0895-7177(03)90042-2 DOI: https://doi.org/10.1016/S0895-7177(03)90042-2

B. Ibrahim and E. Ibikli, Inverse theorems for Bernstein-Chlodowsky type polynomials, J. Math. Kyoto Univ., 46 (2006) 1, pp. 21-29. DOI: https://doi.org/10.1215/kjm/1250281794

S.M. Mazhar and V. Totik, Approximation by modified Szasz operators, Acta Sci. Math., 49 (1985), pp. 257-269.

G.M. Mirakjan, Approximation of continuous functions with the aid of polynomials e−nx∑mk=0Ck,nχk, Dokl. Akad. Nauk SSSR, 31 (1941), pp. 201-205.

A. Sahai and G. Prasad, On simultaneous approximation by modified Lupas operators, J. Approx. Theory, 45 (1985), 122-128, http://dx.doi.org/10.1016/0021-9045(85)90039-5 DOI: https://doi.org/10.1016/0021-9045(85)90039-5

O. Szasz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards Sect. B. 45 (1950), pp. 239-245. DOI: https://doi.org/10.6028/jres.045.024

D.D. Stancu, Asupra unei generaliz ̆ari a polinoamelor lui Bernstein, Studia Universitatis Babes-Bolyai, 14 (1969) 2, pp. 31-45 (in Romanian).

V. Totik, V., Uniform approximation by Szasz-Mirakjan type operators, Acta Math. Hungar. 41 (1983), pp. 291-307, http://dx.doi.org/10.1007/BF01961317 DOI: https://doi.org/10.1007/BF01961317

B. Della Vecchia, G. Mastroianni and J. Szabados, Weighted approximation of functions by Szasz-Mirakyan-type operators, Acta Math. Hungar.,111 (2006), pp. 325-345, http://dx.doi.org/10.1007/s10474-006-0057-1 DOI: https://doi.org/10.1007/s10474-006-0057-1

B. Della Vecchia, G. Mastroianni and J. Szabados, A weighted generalization of Szasz-Mirakyan and Butzer operators, Mediterr. J. Math., 12 (2015) no. 2, pp. 433-454, http://dx.doi.org/10.1007/s00009-014-0413-2 DOI: https://doi.org/10.1007/s00009-014-0413-2

V.N. Mishra, R.B. Gandhi and F. Nasaireh, Simultaneous approximation by Szasz-Mirakjan-Durrmeyer-type operators, Boll. Unione Mat. Ital., 8 (2016) 4, pp 297-305, http://dx.doi.org/10.1007/s40574-015-0045-x DOI: https://doi.org/10.1007/s40574-015-0045-x

V.N. Mishra and R.B. Gandhi, Simultaneous approximation by Szasz-Mirakjan-Stancu-Durrmeyer type operators , Periodica Mathematica Hungarica, 2016, http://dx.doi.org/10.1007/s10998-016-0145-0 DOI: https://doi.org/10.1007/s10998-016-0145-0

V.N. Mishra, H.H. Khan, K. Khatri and L.N. Mishra, Hypergeometric representation for Baskakov-Durrmeyer-Stancu type operators, Bulletin of Mathematical Analysis and Applications, 5 (2013) 3, pp. 18-26.

V.N. Mishra, K. Khatri, L.N. Mishra and Deepmala, nverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and Applications 2013, 2013:586, http://dx.doi.org/10.1186/1029-242X-2013-586 DOI: https://doi.org/10.1186/1029-242X-2013-586

V.N. Mishra, K. Khatri and L.N. Mishra, Some approximation properties of q-Baskakov-Beta-Stancu type operators, Journal of Calculus of Variations, Volume 2013, Article ID 814824, 8 pp., http://dx.doi.org/10.1155/2013/814824 DOI: https://doi.org/10.1155/2013/814824

V.N. Mishra, K. Khatri and L.N. Mishra, Statistical approximation by Kantorovich type discrete q-beta operators, Advances in Difference Equations 2013, 2013:345, http://dx.doi.org/10.1186/10.1186/1687-1847-2013-345 DOI: https://doi.org/10.1186/1687-1847-2013-345

V.N. Mishra, P. Sharma and L.N. Mishra, On statistical approximation properties of q-Baskakov-Szasz-Stancu operators , Journal of Egyptian Mathematical Society, 24 (2016) 3, pp. 396-401, http://dx.doi.org/10.1016/j.joems.2015.07.005 DOI: https://doi.org/10.1016/j.joems.2015.07.005

A. Wafi, N. Rao and Deepmala Rai, Approximation properties by generalized-Baskakov Kantorovich-Stancu type operators, Appl. Math. Inf. Sci. Lett., 4 (2016) 3, pp. 111-118, http://dx.doi.org/10.18576/amisl/040303 DOI: https://doi.org/10.18576/amisl/040303

D.X. Zhou, Weighted approximation by Szasz-Mirakjan operators, J. Approx. Theory 76 (1994), pp. 393-402, http://dx.doi.org/10.1006/jath.1994.1025 DOI: https://doi.org/10.1006/jath.1994.1025

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Published

2016-09-19

How to Cite

Mishra, V. N., Gandhi, R. B., & Mohapatra, R. N. (2016). A summation-integral type modification of Szasz-Mirakjan-Stancu operators. J. Numer. Anal. Approx. Theory, 45(1), 27–36. https://doi.org/10.33993/jnaat451-1075

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