4,422 research outputs found

    Univariate interpolation by exponential functions and gaussian RBFs for generic sets of nodes

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    We consider interpolation of univariate functions on arbitrary sets of nodes by Gaussian radial basis functions or by exponential functions. We derive closed-form expressions for the interpolation error based on the Harish-Chandra-Itzykson-Zuber formula. We then prove the exponential convergence of interpolation for functions analytic in a sufficiently large domain. As an application, we prove the global exponential convergence of optimization by expected improvement for such functions.Comment: Some stylistic improvements and added references following feedback from the reviewer

    Effects of Smoothing Functions in Cosmological Counts-in-Cells Analysis

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    A method of counts-in-cells analysis of galaxy distribution is investigated with arbitrary smoothing functions in obtaining the galaxy counts. We explore the possiblity of optimizing the smoothing function, considering a series of mm-weight Epanechnikov kernels. The popular top-hat and Gaussian smoothing functions are two special cases in this series. In this paper, we mainly consider the second moments of counts-in-cells as a first step. We analytically derive the covariance matrix among different smoothing scales of cells, taking into account possible overlaps between cells. We find that the Epanechnikov kernel of m=1m=1 is better than top-hat and Gaussian smoothing functions in estimating cosmological parameters. As an example, we estimate expected parameter bounds which comes only from the analysis of second moments of galaxy distributions in a survey which is similar to the Sloan Digital Sky Survey.Comment: 33 pages, 10 figures, accepted for publication in PASJ (Vol.59, No.1 in press
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