Compactly Supported, Piecewise Polyharmonic Radial Functions with Prescribed Regularity

A compactly supported radially symmetric function Φ : R → R is said to have Sobolev regularity k if there exist constants B ≥ A > 0 such that the Fourier transform of Φ satisfies A(1 + ‖ω‖) ≤ b Φ(ω) ≤ B(1 + ‖ω‖), ω ∈ R. Such functions are useful in radial basis function methods because the resulting native space will correspond to the Sobolev space W k 2 (R). 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 R of the same parity.