A characterisation of the rate of approximation of Kantorovich sampling operators in weighted variable exponent Lebesgue spaces

We establish a direct and a matching two-term converse estimate by a K-functional and moduli of smoothness for the rate of approximation by generalised Kantorovich sampling operators in weighted variable exponent Lebesgue spaces. They yield the saturation property and class of these operators. The weight is power-type with nonpositive exponents at infinity. We obtain an embedding inequality in weighted variable exponent Lebesgue spaces. We establish main properties of the moduli of smoothness. We demonstrate the general results on a sampling operator of that type whose kernel is supported on an arbitrarily fixed finite interval and which provides a rate of approximation of any power-type order given in advance.