2-SCALE DIFFERENCE-EQUATIONS .2. LOCAL REGULARITY, INFINITE PRODUCTS OF MATRICES AND FRACTALS

This paper studies solutions of the functional equation f(x) = SIGMA(n=0)N c(n)f(kx - n) where k greater-than-or-equal-to 2 is an integer, and SIGMA(n=0)N c(n) = 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 {c(n)} for a continuous nonzero L1-solution to exist. Additional conditions sufficient to guarantee f is-an-element-of C(r) 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 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. For certain f(x) a hierarchy of fractal sets in R corresponding to different Holder exponents of continuity for f(x) is described.