Approximation by Kantorovich-type operators in general Banach spaces

This paper investigates the approximation properties of linear Kantorovich-type sampling operators in the setting of general Banach lattices $X$ on the torus $\mathbb{T}$ and the real line $\mathbb{R}$. Under a natural assumption on the uniform boundedness of the Steklov averaging operators, we establish direct and inverse approximation estimates, as well as strong converse inequalities. Our framework does not require the underlying spaces to be translation-invariant, thereby covering a wide variety of classes and extending the estimates for Kantorovich-type operators previously known primarily for Lebesgue spaces $L_p$.