Kantorovich-Bernstein α-fractal function in 𝓛P spaces

Abstract

Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein α-fractal operator in the Lebesgue space 𝓛p(I), 1 ≤ p ≤ ∞. The main aim of this article is to study the convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in 𝓛p(I) spaces and Lipschitz spaces without affecting the non-linearity of the fractal functions. In the first part of this paper, we introduce a new family of self-referential fractal 𝓛p(I) functions from a given function in the same space. The existence of a Schauder basis consisting of self-referential functions in 𝓛p spaces is proven. Further, we derive the fractal analogues of some 𝓛p(I) approximation results, for example, the fractal version of the classical Müntz-Jackson theorem. The one-sided approximation by the Bernstein α-fractal function is developed.