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An Inverse Result of Approximation by Sampling Kantorovich Series
Part of
- Approximations and expansions
- General theory of linear operators
Danilo Costarelli, Gianluca Vinti
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 62 / Issue 1 / February 2019

Published online by Cambridge University Press:

16 October 2018, pp. 265-280
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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.