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

Fast algorithms for interpolation with L-splines for differential operators L of order 4 with constant coefficients