Quartic Gaussian and Inverse-Quartic Gaussian radial basis functions: The importance of a nonnegative Fourier transform

Authors:
John P. Boyd;Philip W. Mccauley
Affiliations:
-;-
Venue:
Computers & Mathematics with Applications
Year:
2013

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Abstract

We catalogue the numerical properties of approximations using two novel types of radial basis functions @f(r). The QG species is a basis of exponentials of quartic argument: f(x)~f^R^B^F(x;@a,h)=@?"j"="1^Na"jexp(-[@a/h]^4(x-x"j)^4) where the x"j are the RBF centers and also the interpolation points. We show that Quartic Gaussian RBFs fail at many discrete values of the shape parameter @a. We show through a detailed analysis that these singularities are directly related to zeros of Q(k), the Fourier Transform of exp(-x^4). If we reverse the roles and take Q(x) as the RBF, all difficulties disappear because these IQG RBFs have a Fourier transform which is nonnegative for all real k. We explain that although the Quartic-Gaussian exp(-x^4) is positive definite in the physics/dynamical systems sense of being zero-free and nonnegative, it lacks the crucial property of being positive definition in the RBF/analysis sense.