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Kantorovich-Bernstein _α-Fractal function in Lp spaces


A.K.B. Chand
Sangita Jha
M.A. Navascués

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 Lp(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 Lp(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 Lp(I) functions from a given function in the same space. The existence of a Schauder basis consisting of selfreferential 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 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.


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eISSN: 1727-933X
print ISSN: 1607-3606