Asymptotic normality of scaling functions

Abstract

The Gaussian function G(x) = 1/√2πe -x2/2, which has been a classical choice for multiscale representation, is the solution of the scaling equation G(x) = ∫ ℝ αG(αx - y)dg(y), x ∈ ℝ, with scale α > 1 and absolutely continuous measure dg(y)= 1/√2π(α 2-1) e -y2/2(α2-1)dy. It is known that the sequence of normalized B-splines (B n), where B n is the solution of the scaling equation φ(x) = Σ j=0 n1/2 n-1( j n)φ(2x - j), x ∈ ℝ, converges uniformly to G. The classical results on normal approximation of binomial distributions and the uniform B-splines are studied in the broader context of normal approximation of probability measures m n, n = 1, 2,... , and the corresponding solutions φ n of the scaling equations φ n(x) = ∫ ℝ αφ n(αx - y)dm n(y), x ∈ ℝ. Various forms of convergence are considered and orders of convergence obtained. A class of probability densities are constructed that converge to the Gaussian function faster than the uniform B-splines. © 2004 Society for Industrial and Applied Mathematics.