Asymptotic normality of scaling functions

The Gaussian function G(x) = 1/root(2π) over bare(-x2/2), which has been a classical choice for multiscale representation, is the solution of the scaling equation G(x) = integral(R) alphaG(alphax - y)dg(y), x is an element of R, with scale alpha > 1 and absolutely continuous measure dg(y) = 1/root(2π(α2 -1) over bare(-y2/2(alpha2- 1)dy). It is known that the sequence of normalized B-splines (B-n), where B-n is the solution of the scaling equation phi(x) = [GRAPHICS] 1/2(n-1) ((n)(j))phi(2x - j), x is an element of 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 phi(n) of the scaling equations phi(n)(x) = integral(R) alphaphi(n)(alphax - y)dm(n)(y), x is an element of 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.