M.J.D. Powell

On thin plate spline interpolation in two and one dimensions

We consider thin plate spline interpolation in two dimensions to values of a function $f$ at an infinite sequence of points $x_j$, $j \!=\! 0,1,2, \ldots$, where $x_0$, $x_1$ and $x_2$ are not collinear, and where $f$ is twice differentiable, the integrals of squares of its second derivatives being finite. For $k \!\geq\! 2$, let $s_k$ be the thin plate spline interpolant to $f( x_j )$, $j \!=\! 0,1,\ldots,k$. We find that the sequence $s_k$, $k \!=\! 2,3,\ldots$, converges pointwise uniformly to a function $s_* (x)$, $x \!\in\! {\cal R}^2$. Moreover, if the data points lie on a straight line, then this property holds on the line, so the two-dimensional results become relevant to thin plate spline interpolation in one dimension. We now define $s_k$ by interpolating the values $f(jh)$, $j \!=\! 0,1,\ldots,k$, where $h \!=\! 1/k$. It is known that, if $f$ is smooth and $x$ is well inside the interval $[0,1]$, then $| s_k (x) \!-\! f(x) |$ is of magnitude $h^3$, but numerical experiments show that errors of magnitude $h^{3/2}$ are usual near the ends of the interval, even if the support of $f$ is strictly inside $[0,1]$. We investigate this observation, using the results that are mentioned above.