An inverse result of approximation by sampling Kantorovich series

TL;DR

This paper establishes a saturation theorem for sampling Kantorovich operators, showing the maximum approximation order is one unless the function is constant, using a representation formula linked to generalized sampling operators.

Contribution

It introduces a new inverse approximation result (saturation theorem) for sampling Kantorovich operators, connecting their approximation limits to function constancy.

Findings

Maximum approximation order is one for these operators.

If higher order is achieved, the function must be constant.

Representation formula relates Kantorovich series to generalized sampling operators.

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, here 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 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 P.L. Butzer. At the end, some other applications of such representation formula are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.