Asymptotic normality of scaling functions

Timothy Goodman, Louis H. Y. Chen, S. L. Lee

    Research output: Contribution to journalArticlepeer-review

    10 Citations (Scopus)


    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.
    Original languageEnglish
    Pages (from-to)323-346
    Number of pages24
    JournalApplied and Computational Harmonic Analysis
    Issue number1
    Publication statusPublished - 2004


    • Normal approximation
    • Probability measures
    • Scaling functions
    • Uniform \it B-splines
    • Asymptotic normality


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