Compactly supported radial functions and the Strang-Fix condition

摘要

This paper proves some negative results concerning continuous d-variate positive definite radial functions Ф(·) = Ф(‖·‖2) for space dimensions d ≥2: •• If Φ is compactly supported, it cannot satisfy a Strang-Fix condition of positive order.•• If Ф(0) is well defined and nonzero, then Φ does not satisfy a refinement equation.•• These two results also hold for finite linear combinations of integer translates of a compactly supported Φ.•• If Φ is conditionally positive definite on Rd with a generalized Fourier transform Ф that has a Laurent expansion around zero, and if Φ satisfies a refinement equation, then Φ is a polyharmonic spline.