Journal Article

Asymptotic Normality of Scaling Functions

Chen, Louis H. Y., Goodman, Tim N. T., Lee, S. L.

2004

Overview

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) = ınt_ R G( x - y) dg(y), xın R, \\] with scale$\\alpha >1$and absolutely continuous measure \\[ dg (y) = 12 (^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$$ \\phi(x) = \\sum_{j=0}^n \\frac{1}{2^{n-1}} \\binom{n}{j} \\phi(2 x -j), \\quad xın {\\mathbb R}, $$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$ $_n (x) = ınt_ R _n ( x -y) dm_n (y), xın 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.

• Publisher: Society for Industrial and Applied Mathematics
• Subject: Algorithms / Applied mathematics / Approximation / Approximations and expansions / Binomial distribution / Exact sciences and technology / Fourier analysis / Fourier transforms / Mathematical analysis / Mathematics / Numerical analysis / Numerical analysis. Scientific computation / Random variables / Sciences and techniques of general use