Equivalent Kernels for Smoothing Splines

In the study of smoothing spline estimators, some convolution-kernellike properties of the Green’s function for an appropriate boundary value problem, depending on the design density, are needed. For the uniform density, the Green’s function can be computed more or less explicitly. Then, integral equation methods are brought to bear to establish the kernel-like properties of said Green’s function. We briefly survey how the Green’s function arises in spline smoothing as the equivalent kernel, the reproducing kernel of a suitable Hilbert space, and as the Green’s function for the Euler equations of a semicontinuous version of the spline smoothing problem.