Nonlinear Fractal Histopolation Function

Abstract: A novel method for constructing a nonlinear fractal histopolation function associated with a given histogram is introduced in this paper. In contrast to classical fractal interpolation methods, which produce continuous and interpolatory functions, the proposed approach constructs a bounded, Riemann integrable function that is not necessarily continuous but preserves the area of a given histogram. An iterated function system based on Rakotch contractions- a generalisation of Banach contractions- is utilised, thereby extending the theoretical framework for fractal histopolation. Unlike previous formulations, the proposed construction of nonlinear fractal functions allows vertical scaling factors greater than one. The conditions for the nonlinear fractal function to be a solution for the histopolation problem are derived.