9 - Fractal surfaces

Abstract

This chapter deals with Rm-valued multivariate continuous fractal functions f:X⊆Rn→Rm and some of their properties. The existence and construction of these multivariate functions is already implicitly contained in Theorems 71–73Theorem 71Theorem 72Theorem 73; simply choose X to be a nonempty compact subset of Rn and Y:=Rm. Here, however, the issues have to be readdressed. There are two reasons for this: firstly, the more complex geometry is hidden within the construction and needs to be investigated more closely; secondly, the graphs of these Rm-valued multivariate continuous fractal functions, the so-called fractal surfaces, can be used to construct wavelet bases in Rn. This construction is based on results from the theory of Coxeter and affine reflection groups, and certain issues involving the geometry of fractal surfaces need to be clarified. Chapter 10 will deal exclusively with this last question.