### Some properties of thin plate spline interpolation

## Abstract

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.