A note on the limited stability of surface spline interpolation

Given a finite subset Ξ⊂Rd and data f|Ξ, the surface spline interpolant to the data f|Ξ is a function s which minimizes a certain seminorm subject to the interpolation conditions s|Ξ=f|Ξ. It is known that surface spline interpolation is stable on the Sobolev space Wm in the sense that ∥s∥L∞(Ω)⩽const∥f∥Wm, where m is an integer parameter which specifies the surface spline. In this note we show that surface spline interpolation is not stable on Wγ whenever .