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Approximation of functions by a new class of linear operators

Published online by Cambridge University Press:  09 April 2009

G. C. Jain
G. C. Jain
Affiliation:
University of Calgary Alberta, Canada
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Various extensions and generalizations of Bernstein polynomials have been considered among others by Szasz [13], Meyer-Konig and Zeller [8], Cheney and Sharma [1], Jakimovski and Leviatan [4], Stancu [12], Pethe and Jain [11]. Bernstein polynomials are based on binomial and negative binomial distributions. Szasz and Mirakyan [9] have defined another operator with the help of the Poisson distribution. The operator has approximation properties similar to those of Bernstein operators. Meir and Sharma [7] and Jam and Pethe [3] deal with generalizations of Szasz-Mirakyan operator. As another generalization, we define in this paper a new operator with the help of a Poisson type distribution, consider its convergence properties and give its degree of approximation. The results for the Szasz-Mirakyan operator can easily be obtained from our operator as a particular case.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 3 , May 1972 , pp. 271 - 276
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Cheney, E. W. and Sharma, A., ‘On a Generalization of Bernstein Polynomials’, Rev. Mat. Univ. Parma (2) 5 (1964), 77–84.Google Scholar
[2]Consul, P. C. and Jain, G. C., ‘On a Generalized Poisson Distribution’, Submitted (1970).Google Scholar
[3]Jain, G. C. and Pethe, S. P., ‘On a Generalization of Szasz-Mirakyan operator’, Mathematica (Cluj) (2) 35 (1970), 313–318.Google Scholar
[4]Jakimovski, A. and Leviatan, D., ‘Generalized Bernstein Polynomials’, Math. Z. 93 (1966), 411–426.CrossRefGoogle Scholar
[5]Jensen, J. L. W. V., ‘Sur une identité, d'Abel et sur d'autres formules analogues’, Acta Math. 26, (1902), 307–318.CrossRefGoogle Scholar
[6]Korovkin, P. P., Linear Operators and Approximation Theory (translated from Russian edition of 1959, Delhi, 1960.)Google Scholar
[7]Meir, A. and Sharma, A., ‘Approximation Methods by Polynomials and Power Series’, Koninkl. Nederl. Akademie Van Wetenschappen. Amsterdam, Reprinted from Proceedings, Series A, 70, No. 4 andCrossRefGoogle Scholar
Indag. Math. 29 (1967), 77–84.Google Scholar
[8]Meyer-König, W. and Zeller, K., ‘Bernsteinsche Potenzreihen’, Studia Math. 19 (1960), 89–94.CrossRefGoogle Scholar
[9]Mirakyan, G., ‘Approximation des Fonctions Continues au Moyen de Polynomes de la Forme’, Dokl. Akad. Nauk. SSSR 31 (1941), 201–205.Google Scholar
[10]Müller, M., Die Folge der Gamma Operatoren, Dissertation, Stuttgart, 1967).Google Scholar
[11]Pethe, S. P. and Jain, G. C., ‘Approximation of functions by Bernstein type operator’, Canad. Math. Bull., to appear.Google Scholar
[12]Stancu, D. D., ‘Approximation of functions by a New Class of Linear Polynomial Operators’, Revue Roumaine de Mathématique Pures et Appliquées 8 (XIII) (1968), 1173–1994.Google Scholar
[13]Szasz, O., ‘Generalization of S. Bernstein's Polynomials to the Infinite Interval’, J. of Research of the Nat. Bur. of Standards, 45, (1950) 239–245;CrossRefGoogle Scholar
Collected Mathematical Works (Cincinnati 1955, 1401–1407).Google Scholar