Construction and Box-counting Dimension of the Edelstein Hidden Variable Fractal Interpolation Function
Aiswarya T, Srijanani Anurag Prasad
TL;DR
This work constructs a hidden-variable fractal interpolation function using Edelstein contractions within an iterated function system built from a finite data set, yielding a vector-valued fractal interpolant. A Read-Bajraktarević-type fixed-point framework is employed to obtain a fixed point whose first component is the Edelstein HVFIF. The paper proves Hölder continuity of the first component with exponent $\alpha$ under suitable choices of parameters, and shows that the graph's upper box-counting dimension satisfies $\overline{dim}_B(G(f_1)) \le 2-\alpha$. These results advance the understanding of smoothness and fractal dimension for hidden-variable FIFs and enhance their applicability to data exhibiting self-similarity in a hidden-variable fractal framework.
Abstract
This paper presents the construction of a hidden variable fractal interpolation function using Edelstein contractions in an iterated function system based on a finite collection of data points. The approach incorporates an iterated function system where variable functions act as vertical scaling factors leading to a generalised vector-valued fractal interpolation function. Furthermore, the paper rigorously examines the smoothness of the constructed function and establishes an upper bound for the box-counting dimension of its graph.