The uniform convergence of thin plate spline interpolation in two dimensions

Abstract

Let f be a function from R 2 to R that has square integrable second derivatives and let s be the thin plate spline interpolant to f at the points {v i : i = 1, 2, . . . , n} in R 2 . We seek bounds on the error |f (x)−s(x)| when x is in the convex hull of the interpolation points or when x is close to at least one of the interpolation points but need not be in the convex hull. We find, for example, that, if x is inside a triangle whose vertices are any three of the interpolation points, then |f (x)−s(x)| is bounded above by a multiple of h, where h is the length of the longest side of the triangle and where the multiplier is independent of the interpolation points. Further, if D is any bounded set in R 2 that is not a subset of a single straight line, then we prove that a sequence of thin plate spline interpolants converges to f uniformly on D . Specifically, we require h → 0, where h is now the least upper bound on the numbers {d(x, V ) : x ∈ D } and where d(x, V ), x ∈ R 2 , is the least Euclidean distance from x to an interpolation point. Our method of analysis applies integration by parts and the Cauchy-Schwarz inequality to the scalar product between second derivatives that occurs in the variational calculation of thin plate spline interpolation.

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