  This paper considers the development of spatially adaptive smoothing splines
for the estimation of a regression function with non-homogeneous smoothness
across the domain. Two challenging issues that arise in this context are the
evaluation of the equivalent kernel and the determination of a local penalty.
The roughness penalty is a function of the design points in order to
accommodate local behavior of the regression function. It is shown that the
spatially adaptive smoothing spline estimator is approximately a kernel
estimator. The resulting equivalent kernel is spatially dependent. The
equivalent kernels for traditional smoothing splines are a special case of this
general solution. With the aid of the Green's function for a two-point boundary
value problem, the explicit forms of the asymptotic mean and variance are
obtained for any interior point. Thus, the optimal roughness penalty function
is obtained by approximately minimizing the asymptotic integrated mean square
error. Simulation results and an application illustrate the performance of the
proposed estimator.
