Kantorovich-Bernstein _α-Fractal function in Lp spaces

Abstract

Fractal interpolation functions are xed points of contraction maps on suitable function spaces. In this paper, we introduce the  Kantorovich-Bernstein -fractal operator in the Lebesgue space L^p(I); 1 ≤ p ≤ 1. The main aim of this article is to study the  convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in L^p(I) spaces and Lipschitz spaces without aecting the non-linearity of the fractal functions. In the rst part of this paper, we introduce a new family of self-referential fractal L^p(I) functions from a given function in the same space. The existence of a Schauder basis consisting of selfreferential functions in L^p spaces is proven. Further, we derive the fractal analogues of some L^p(I) approximation results, for  example, the fractal version of the classical Muntz-Jackson theorem. The one-sided approximation by the Bernstein -fractal function is developed. Mathematics Subject Classication (2010): 28A80, 41A25, 47A09, 47A05, 58C07.

Key words: Fractal interpolation, -fractal operator, Bernstein-Kantorovich polynomial, function spaces, Schauder basis.