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 γ < m - 1/2.

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