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.