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Sharp estimates of approximation by some positive linear operators

Published online by Cambridge University Press:  17 April 2009

Ashok Sahai  and
Govind Prasad
Ashok Sahai
Affiliation:
Department of Mathematics, University of Roorkee, Roorkee, U.P. 247667, India.
Govind Prasad
Affiliation:
Department of Mathematics, University of Roorkee, Roorkee, U.P. 247667, India.
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Abstract

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Recently, Varshney and Singh [Rend. Mat. (6) 2 (1982), 219–225] have given sharper quantitative estimates of convergence for Bernstein polynomials, Szasz and Meyer-Konig-Zeller operators. We have achieved improvement over these estimates by taking moments of higher order. For example, in case of the Meyer-Konig-Zeller operator, they gave the following estimate

wherein ∥·∥ stands for sup norm. We have improved this result to

We may remark here that for this modulus of continuity ) our result cannot be sharpened further by taking higher order moments.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 29 , Issue 1 , February 1984 , pp. 13 - 18
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Rathore, R.K.S., “Lipschitz-Nikolski constants and asymptotic simultaneous approximation of the M n-operators”, Aequationes Math. 18 (1978), 206–217.CrossRefGoogle Scholar
[2]Varshney, Om P. and Singh, S.P., “On degree of approximation by positive linear operators”, Rend. Mat. (7) 2 (1982), 219–225.Google Scholar