The uniform convergence of thin plate spline interpolation in two dimensions

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.