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Mollification formulas and implicit smoothing

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Abstract

This paper develops some mollification formulas involving convolutions between popular radial basis function (RBF) basic functions Φ, and suitable mollifiers. Polyharmonic splines, scaled Bessel kernels (Matern functions) and compactly supported basic functions are considered. A typical result is that in ℛd the convolution of |{•}|β and (•2+c 2)−(β+2d)/2 is the generalized multiquadric (•2+c 2)β/2 up to a multiplicative constant. The constant depends on c>0, β, where ℜ(β)>−d, and d. An application which motivated the development of the formulas is a technique called implicit smoothing. This computationally efficient technique smooths a previously obtained RBF fit by replacing the basic function Φ with a smoother version Ψ during evaluation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

    R. K. Beatson & H. Q. Bui

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  1. R. K. Beatson
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  2. H. Q. Bui
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Communicated by R. Schaback

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Beatson, R.K., Bui, H.Q. Mollification formulas and implicit smoothing. Adv Comput Math 27, 125–149 (2007). https://doi.org/10.1007/s10444-005-7512-3

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  • DOI: https://doi.org/10.1007/s10444-005-7512-3

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