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

This paper studies solutions of the functional equation f(x) = sum from N to n=0 c sub n f(kx - n) where k >= 2 is an integer, and sum from N to n=0 c sub n = k. Part I showed that equations of this type have at most one L sup 1 -solution, up to a multiplicative constant, which necessarily has compact support in [O, N over k-1]. This paper gives a time-domain representation for such a function f(x) (if is exists) in terms of infinite products of matrices (that vary as x varies). We give new sufficient conditions on {c sub n} for a continuous nonzero L sup 1 -solution to exist. We also give additional conditions sufficient to guarantee f epsilon C sup r. We use the infinite matrix product representation to bound from below the degree of regularity of such an L sup 1 -solution, and to estimate the Holder 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. We describe for certain f(x) a hierarchy of fractal sets in R corresponding to different Holder exponents of continuity for f(x).