Asymptotic Normality of Scaling Functions

The Gaussian function G(x) = 1 p 21⁄4 e¡x 2=2; which has been a classical choice for multiscale representation, is the solution of the scaling equation G(x) = Z R ®G(®x¡ y)dg(y); x 2 R; with scale ® > 1 and absolutely continuous measure dg(y) = 1 p 21⁄4(®2 ¡ 1) e¡y 2=2(®2¡1)dy: It is known that the sequence of normalized B-splines (Bn); where Bn is the solution of the scaling equation Á(x) = n X j=0 1 2n¡1 μ n j ¶ Á(2x¡ j); x 2 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 Án of the scaling equations Án(x) = Z R ®Án(®x¡ y)dmn(y); x 2 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.