  The popular cubic smoothing spline estimate of a regression function arises
as the minimizer of the penalized sum of squares $\sum_j(Y_j - {\mu}(t_j))^2 +
{\lambda}\int_a^b [{\mu}"(t)]^2 dt$, where the data are $t_j,Y_j$, $j=1,...,
n$. The minimization is taken over an infinite-dimensional function space, the
space of all functions with square integrable second derivatives. But the
calculations can be carried out in a finite-dimensional space. The reduction
from minimizing over an infinite dimensional space to minimizing over a finite
dimensional space occurs for more general objective functions: the data may be
related to the function ${\mu}$ in another way, the sum of squares may be
replaced by a more suitable expression, or the penalty, $\int_a^b [{\mu}"(t)]^2
dt$, might take a different form. This paper reviews the Reproducing Kernel
Hilbert Space structure that provides a finite-dimensional solution for a
general minimization problem. Particular attention is paid to penalties based
on linear differential operators. In this case, one can sometimes easily
calculate the minimizer explicitly, using Green's functions.
