Asymptotic theory of finite dimensional normed spaces
Asymptotic theory of finite dimensional normed spaces
Eigenvalues of Euclidean distance matrices
Journal of Approximation Theory
A note on the local stability of translates of radical basis functions
Journal of Approximation Theory
Norm estimates for inverses of Toeplitz distance matrices
Journal of Approximation Theory
Convolution operators for radial basis approximation
SIAM Journal on Mathematical Analysis
Radial Basis Functions
On Spherical Averages of Radial Basis Functions
Foundations of Computational Mathematics
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A radial basis function approximation takes the form $$s(x)=\sum_{k=1}^na_k\phi(x-b_k),\quad x\in {\mathbb{R}}^d,$$ where the coefficients a 1,驴,a n are real numbers, the centres b 1,驴,b n are distinct points in 驴 d , and the function 驴:驴 d 驴驴 is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which 驴 is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure μ for which the convolution 驴=μ 驴 is a function of compact support, and when 驴 is polyharmonic. The novelty of this construction is its use of the Paley---Wiener theorem to identify compact support via analysis of the Fourier transform of the new kernel 驴, so providing a new form of kernel engineering.