SELF-SIMILAR MEASURES AND THEIR FOURIER-TRANSFORMS .2.

A self-similar measure on Rn was defined by Hutchinson to be a probability measure satisfying [GRAPHICS] where S(j)x = rho(j)R(j)x + b(j) is a contractive similarity (0 < rho(j) < 1 , R(j) orthogonal) and the weights a(j) satisfy 0 < a(j) < 1 , SIGMA(j=1)m a(j) = 1. By analogy, we define a self-similar distribution by the same identity (*) but allowing the weights aj to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to (*) among distributions of compact support, and show that the space of such solutions is always finite dimensional. If F denotes the Fourier transformation of a self-similar distribution of compact support, let H(R) = 1/R(n-beta) integral(\x\ less-than-or-equal-to R)\F(x)\2 dx, where beta is defined by the equation SIGMA(j=1)m rho(j)-beta\a(j)\2 = 1. If rho(j)upsilonj = rho for some fixed rho and upsilon(j) positive integers we say the {rho(j)} are exponentially commensurable. In this case we prove (under some additional hypotheses) that H(R) is asymptotic (in a suitable sense) to a bounded function H(R) that is bounded away from zero and periodic in the sense that H(rhoR) = H(R) for all R > 0. If the {rho(j)} are exponentially incommensurable then lim(R-->infinity) H(R) exists and is nonzero.