An Inverse Result of Approximation by Sampling Kantorovich Series

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 a...

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Bibliographic Details
Published inProceedings of the Edinburgh Mathematical Society Vol. 62; no. 1; pp. 265 - 280
Main Authors Costarelli, Danilo, Vinti, Gianluca
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.02.2019
Subjects
Approximation
Continuity (mathematics)
Mathematical analysis
Operators (mathematics)
Representations
Sampling
saturation theorem
central B-splines
sampling Kantorovich series
41A30
order of approximation
47A58
generalized sampling operators
inverse results
Primary 41A25
41A05
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Summary: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.
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content type line 14
ISSN:0013-0915
1464-3839
DOI:10.1017/S0013091518000342