Moments Of The Weighted Cantor Measures

Based on the seminalwork of Hutchinson, we investigate properties of alpha-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, alpha is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure mu(alpha) is the unique Borel probability measure on [0, 1] satisfying mu(alpha) (E) = Sigma(N-1)(n=0) alpha(n)mu(alpha) (phi(-1)(n)(E)) where phi(n) : x bar right arrow (x + n)/N. In Sections 1 and 2 we examine several general properties of the measure mu(alpha) and the associated Legendre polynomials in L-mu alpha(2) [0, 1]. In Section 3, we (1) compute the Laplacian and moment generating function of mu(alpha), (2) characterize precisely when the moments I-m = integral([0,1]) chi(m) d mu(alpha) exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first m moments within uniform error epsilon in O((log log(1/epsilon)) . m log m). We also state analogous results in the natural case where alpha is palindromic for the measure nu(alpha) attained by shifting mu(alpha) to [-1/2, 1/2].