  In the classical theory of cubic interpolation splines there exists an
algorithm which works with only $O\left( n\right)$ arithmetic operations. Also,
the smoothing cubic splines may be computed via the algorithm of Reinsch which
reduces their computation to interpolation cubic splines and also performs with
$O\left( n\right)$ arithmetic operations. In this paper it is shown that many
features of the polynomial cubic spline setting carry over to the larger class
of $L$-splines where $L$ is a linear differential operator of order $4$ with
constant coefficients. Criteria are given such that the associated matrix $R$
is strictly diagonally dominant which implies the existence of a fast algorithm
for interpolation.
