Mollification formulas and implicit smoothing

Authors: Beatson, R.; Bui, H.

Source: Advances in Computational Mathematics, Volume 27, Number 2, August 2007 , pp. 125-149(25)

Publisher: Springer

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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 ^d 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.

Keywords: mollification formulas; radial basis functions; implicit smoothing; data smoothing; 41A30; 65D10

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10444-005-7512-3

Publication date: 2007-08-01