Fractal Interpolation Functions from $R^n$ into $R^m$ and their Projections

Abstract

We construct fractal interpolation functions from ${\mathbb{R}}^n\to {\mathbb{R}}^m$, and consider the projections of their graphs onto ${\mathbb{R}}^n\times {\mathbb{R}}^m$. Since these projections still depend continuously on all the variables we refer to them as hidden variable fractal interpolation surfaces. The hidden variable fractal interpolation surfaces carry additional free parameters and are thus more general than, for instance, the fractal surfaces defined earlier by the authors. These free parameters may prove useful in approximation-theoretic considerations. A formula for the box dimension of a hidden variable fractal interpolation surface is presented.. This dimension parameter could be used to distinguish different textures on natural surfaces.