A fractal operator associated to bivariate fractal interpolation functions

TL;DR

This paper introduces a new bivariate fractal operator based on fractal interpolation surfaces, extending univariate fractal operators to two variables and exploring their properties and approximation capabilities.

Contribution

It develops a bounded linear bivariate fractal operator analogous to the univariate $\alpha$-fractal operator, extending it to Lp-spaces and analyzing its approximation properties.

Findings

Defined a new bivariate fractal operator

Extended the operator to Lp-spaces for 1 ≤ p < ∞

Discussed approximation properties of the fractal functions

Abstract

A general framework to construct fractal interpolation surfaces (FISs) on rectangular grids was presented and bilinear FIS was deduced by Ruan and Xu [Bull. Aust. Math. Soc. 91(3), 2015, pp. 435-446]. From the view point of operator theory and the stand point of developing some approximation aspects, we revisit the aforementioned construction to obtain a fractal analogue of a prescribed continuous function defined on a rectangular region in $R^2$. This approach leads to a bounded linear operator analogous to the so-called $\alpha$-fractal operator associated with the univariate fractal interpolation function. Several elementary properties of this bivariate fractal operator are reported. We extend the fractal operator to the Lp-spaces for $1 \le p < \infty$. Some approximation aspects of the bivariate continuous fractal functions are also discussed.