  This paper addresses asymptotic properties of general penalized spline
estimators with an arbitrary B-spline degree and an arbitrary order difference
penalty. The estimator is approximated by a solution of a linear differential
equation subject to suitable boundary conditions. It is shown that, in certain
sense, the penalized smoothing corresponds approximately to smoothing by the
kernel method. The equivalent kernels for both inner points and boundary points
are obtained with the help of Green's functions of the differential equation.
Further, the asymptotic normality is established for the estimator at interior
points. It is shown that the convergence rate is independent of the degree of
the splines, and the number of knots does not affect the asymptotic
distribution, provided that it tends to infinity fast enough.
