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An Inverse Result of Approximation by Sampling Kantorovich Series

Part of: Approximations and expansions General theory of linear operators

Published online by Cambridge University Press:  16 October 2018

Danilo Costarelli Open the ORCID record for Danilo Costarelli [Opens in a new window]  and
Gianluca Vinti
Danilo Costarelli*
Affiliation:
Department of Mathematics and Computer Science, University of Perugia, 1 Via Vanvitelli, 06123 Perugia, Italy (danilo.costarelli@unipg.it; gianluca.vinti@unipg.it)
Gianluca Vinti
Affiliation:
Department of Mathematics and Computer Science, University of Perugia, 1 Via Vanvitelli, 06123 Perugia, Italy (danilo.costarelli@unipg.it; gianluca.vinti@unipg.it)
*
*Corresponding author.
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Abstract

In the present paper, an inverse result of approximation, i.e. a saturation theorem for the sampling Kantorovich operators, is derived in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns out to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by Butzer. At the end, some other applications of such representation formulas are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

Keywords

inverse resultssampling Kantorovich seriesorder of approximationcentral B-splinesgeneralized sampling operatorssaturation theorem

MSC classification

Primary: 41A25: Rate of convergence, degree of approximation
Secondary: 41A05: Interpolation 41A30: Approximation by other special function classes 47A58: Operator approximation theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 265 - 280
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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