doi:10.2989/16073606.2019.1572664engChand, A.K.B.Jha, S.Navascués, M.A.Kantorovich-Bernstein a-fractal function in LP spacesART-2019-111533Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein a-fractal operator in the Lebesgue space Lp(I), 1 = p = 8. 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 a-fractal function is developed.2019http://zaguan.unizar.es/record/8844310.2989/16073606.2019.1572664http://zaguan.unizar.es/record/88443oai:zaguan.unizar.es:88443Quaestiones Mathematicae 43, 2 (2019), 227 - 241All rights reservedhttp://www.europeana.eu/rights/rr-f/info:eu-repo/semantics/openAccess