# The uniform convergence of thin plate spline interpolation in two dimensions

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## Summary.

Let \(f\) be a function from \({\cal R}^2\) to \({\cal R}\) that has square integrable second derivatives and let \(s\) be the thin plate spline interpolant to \(f\) at the points \(\{ \underline v_i : i \!=\! 1,2,\ldots,n \}\) in\({\cal R}^2\) . We seek bounds on the error \(| f(\underline x)-s(\underline x) |\) when \(\underline x\) is in the convex hull of the interpolation points or when \(\underline x\) is close to at least one of the interpolation points but need not be in the convex hull. We find, for example, that, if \(\underline x\) is inside a triangle whose vertices are any three of the interpolation points, then \(| f(\underline x)-s(\underline x) |\) 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\({\cal D}\) is any bounded set in \({\cal R}^2\) 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 \({\cal D}\). Specifically, we require \(h \!\rightarrow\! 0\), where \(h\) is now the least upper bound on the numbers \(\{ d( \underline x, {\cal V} ) : \underline x \!\in\! {\cal D} \}\) and where \(d( \underline x, {\cal V} )\),\(\underline x \!\in\! {\cal R}^2\) , is the least Euclidean distance from \(\underline x\) 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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