Some elementary properties of Bernstein–Kantorovich operators on mixed norm Lebesgue spaces and their implications in fractal approximation

Abstract

On the one hand, the notion of mixed norm spaces has attracted considerable attention in fields such as harmonic analysis and PDE. On the other hand, a particular modification of the Bernstein operator, the so-called Bernstein–Kantorovich operator, has been of special interest for the approximation of the classical Lp-functions. This note has a double purpose. First, we record some elementary approximation properties of the Bernstein–Kantorovich operators on the mixed norm Lebesgue spaces. In the second part, we construct self-referential (fractal) counterparts to the functions belonging to the mixed norm Lebesgue spaces and introduce fractal operators on these spaces. With the help of the Bernstein–Kantorovich operators, we obtain a fractal approximation process on the mixed norm Lebesgue spaces. Furthermore, using the multivariate Haar system, we provide a Schauder basis consisting of self-referential functions for the mixed norm Lebesgue spaces, which we call the Bernstein–Kantorovich fractal Haar system.