On Bivariate Fractal Interpolation for Countable Data and Associated Nonlinear Fractal Operator

TL;DR

This paper introduces a framework for constructing bivariate fractal interpolation surfaces for countably infinite data, and explores the properties of the associated nonlinear fractal operator, bridging fractal interpolation with nonlinear operator theory.

Contribution

It presents a novel method for bivariate fractal interpolation of infinite data sets and analyzes the nonlinear fractal operator's properties, expanding the theoretical understanding.

Findings

Constructed fractal interpolation surfaces for countably infinite data.

Established properties of the nonlinear fractal operator.

Linked fractal interpolation with nonlinear operator theory.

Abstract

We provide a general framework to construct fractal interpolation surfaces (FISs) for a prescribed countably infinite data set on a rectangular grid. Using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operator are established, thereby initiating the interaction of the notion of fractal interpolation with the theory of nonlinear operators.