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

This paper studies solutions of the functional equation \[ f(x) = ∑n = 0^N {c_n f(kx - n),} \] where k 2 is an integer, and ∑n = 0N cₙ = k. Part I showed that equations of this type have at most one L¹-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 \ cₙ \ for a continuous nonzero L¹-solution to exist. Additional conditions sufficient to guarantee f ∈ Cʳ are also given. The infinite matrix product representations is used to bound from below the degree of regularity of such an L¹-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 R corresponding to different Hölder exponents of continuity for $f(x)$ is described.