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Title: Asymptotic normality of scaling functions
Authors: Chen, L.H.Y. 
Goodman, T.N.T.
Lee, S.L. 
Keywords: Asymptotic normality
Normal approximation
Probability measures
Scaling functions
Uniform B-splines
Issue Date: 2005
Citation: Chen, L.H.Y., Goodman, T.N.T., Lee, S.L. (2005). Asymptotic normality of scaling functions. SIAM Journal on Mathematical Analysis 36 (1) : 323-346. ScholarBank@NUS Repository.
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.
Source Title: SIAM Journal on Mathematical Analysis
ISSN: 00361410
DOI: 10.1137/S0036141002406229
Appears in Collections:Staff Publications

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