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Direct and converse inequalities for positive linear operators on the positive semi-axis

Part of: Approximations and expansions Integral transforms, operational calculus

Published online by Cambridge University Press:  09 April 2009

José A. Adell  and
Carmen Sangüesa
José A. Adell
Affiliation:
Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain e-mail: adell@posta.unizar.es, csangues@posta.unizar.es
Carmen Sangüesa
Affiliation:
Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain e-mail: adell@posta.unizar.es, csangues@posta.unizar.es
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Abstract

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We consider positive linear operators of probabilistic type L1f acting on real functions f defined on the positive semi-axis. We deal with the problem of uniform convergence of L1f to f, both in the usual sup-norm and in a uniform Lp type of norm. In both cases, we obtain direct and converse inequalities in terms of a suitable weighted first modulus of smoothness of f. These results are applied to the Baskakov operator and to a gamma operator connected with real Laplace transforms, Poisson mixtures and Weyl fractional derivatives of Laplace transforms.

Keywords

positive linear operatordirect and converse inequalitiesweighted modulus of smoothnessBaskakov operatorgamma operatorLaplace transformPoison mixtureWeyl fractional derivative

MSC classification

Secondary: 44A10: Laplace transform 41A35: Approximation by operators (in particular, by integral operators)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 1 , February 1999 , pp. 90 - 103
Copyright
Copyright © Australian Mathematical Society 1999

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