Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
The missing Wendland functions
Advances in Computational Mathematics
Notes: Minimal degree univariate piecewise polynomials with prescribed Sobolev regularity
Journal of Approximation Theory
Closed form representations for a class of compactly supported radial basis functions
Advances in Computational Mathematics
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In this paper we investigate the generalisation of Wendland's compactly supported radial basis functions to the case where the smoothness parameter is not assumed to be a positive integer or half-integer and the parameter @?, which is chosen to ensure positive definiteness, need not take on the minimal value. We derive sufficient and necessary conditions for the generalised Wendland functions to be positive definite and deduce the native spaces that they generate. We also provide closed form representations for the generalised Wendland functions in the case when the smoothness parameter is an integer and where the parameter @? is any suitable value that ensures positive definiteness, as well as closed form representations for the Fourier transform when the smoothness parameter is a positive integer or half-integer.