Some properties of thin plate spline interpolation

Prof Mike J D Powell
Let the thin plate spline radial basis function method be applied to interpolate values of a smooth function $f(x)$, $x \!\in\! {\cal R}^d$. It is known that, if the data are the values $f(jh)$, $j \in {\cal Z}^d$, where $h$ is the spacing between data points and ${\cal Z}^d$ is the set of points in $d$ dimensions with integer coordinates, then the accuracy of the interpolant is of magnitude $h^{d+2}$. This beautiful result, due to Buhmann, will be explained briefly. We will also survey some recent findings of Bejancu on Lagrange functions in two dimensions when interpolating at the integer points of the half-plane ${\cal Z}^2 \cap \{ x : x_2 \!\geq\! 0 \}$. Most of our attention, however, will be given to the current research of the author on interpolation in one dimension at the points $h {\cal Z} \cap [0,1]$, the purpose of the work being to establish theoretically the apparent deterioration in accuracy at the ends of the range from ${\cal O} ( h^3 )$ to ${\cal O} ( h^{3/2} )$ that has been observed in practice. The analysis includes a study of the Lagrange functions of the semi-infinite grid ${\cal Z} \cap \{ x : x \!\geq\! 0 \}$ in one dimension.
  • Computational Mathematics and Applications Seminar