Asymptotic Normality of Scaling Functions

Abstract

The Gaussian function $G(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2},$ which has been a classical choice for multiscale representation, is the solution of the scaling equation \[ G(x) = \int_{\mathbb R} \alpha G(\alpha x - y) dg(y), \quad x\in {\mathbb R}, \] with scale $\alpha >1$ and absolutely continuous measure \[ dg (y) = \frac{1}{\sqrt{2\pi} (\alpha^2-1)}e^{-y^2/2(\alpha^2-1)} dy. \] It is known that the sequence of normalized B-splines (Bn), where Bn is the solution of the scaling equation $$ \phi(x) = \sum_{j=0}^n \frac{1}{2^{n-1}} \binom{n}{j} \phi(2 x -j), \quad x\in {\mathbb R}, $$ 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 mn, n=1,2,. . . , and the corresponding solutions $\phi_n$ of the scaling equations $ $ \phi_n (x) = \int_{\mathbb R} \alpha \phi_n (\alpha x -y) dm_n (y), \quad x\in {\mathbb 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.