A Fractal Operator Associated with Bivariate Fractal Interpolation Functions on Rectangular Grids

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):435–446, 2015). 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 ${\mathbb {R}}^2$. 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 ${\mathcal {L}}^p$-spaces for $1 \le p < \infty$. Some approximation aspects of the bivariate continuous fractal functions are also discussed.