On kernel engineering via Paley-Wiener.

radial basis function approximation takes the form where the coefficients a,..., a are real numbers, the centres b,..., b are distinct points in ℝ, and the function φ:ℝ→ℝ 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 φ 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 ψ= μ φ is a function of compact support, and when φ 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 ψ, so providing a new form of kernel engineering. [ABSTRACT FROM AUTHOR]