Two-Scale Difference Equations II. Local Regularity, Infinite Products of Matrices and Fractals

Abstract

This paper studies solutions of the functional equation \[ f(x) = \sum_{n = 0}^N {c_n f(kx - n),} \] where $k \geqq 2$ is an integer, and $\sum\nolimits_{n = 0}^N {c_n = k} $. Part I showed that equations of this type have at most one $L^1 $-solution up to a multiplicative constant, which necessarily has compact support in $[0,{N / {k - 1}}]$. This paper gives a time-domain representation for such a function $f(x)$ (if it exists) in terms of infinite products of matrices (that vary as x varies). Sufficient conditions are given on $\{ {c_n } \}$ for a continuous nonzero $L^1 $-solution to exist. Additional conditions sufficient to guarantee $f \in C^r $ are also given. The infinite matrix product representations is used to bound from below the degree of regularity of such an $L^1 $-solution and to estimate the Hölder exponent of continuity of the highest-order well-defined derivative of $f(x)$. Such solutions $f(x)$ are often smoother at some points than others. For certain $f(x)$ a hierarchy of fractal sets in $\mathbb{R}$ corresponding to different Hölder exponents of continuity for $f(x)$ is described.