Compactly Supported, Piecewise Polyharmonic Radial Functions with Prescribed Regularity

Abstract

A compactly supported radially symmetric function $\varPhi :\Bbb{R}^{d}\to \Bbb{R}$ is said to have Sobolev regularity k if there exist constants B≥A>0 such that the Fourier transform of Φ satisfies

$$A\bigl(1+\Vert\omega\Vert ^2\bigr)^{-k} \leq\widehat{\varPhi }(\omega )\leq B\bigl (1+\Vert\omega\Vert ^2\bigr)^{-k},\quad\omega\in \Bbb{R}^d.$$

Such functions are useful in radial basis function methods because the resulting native space will correspond to the Sobolev space $W_{2}^{k}(\Bbb{R}^{d})$. For even dimensions d and integers k≥d/4, we construct piecewise polyharmonic radial functions with Sobolev regularity k. Two families are actually constructed. In the first, the functions have k nontrivial pieces, while in the second, exactly one nontrivial piece. We also explain, in terms of regularity, the effect of restricting Φ to a lower dimensional space $\Bbb{R}^{d-2\ell}$ of the same parity.