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

TL;DR

This paper extends fast cubic spline algorithms to a broader class of L-splines derived from fourth-order linear differential operators with constant coefficients, enabling efficient interpolation.

Contribution

It demonstrates that properties of cubic splines extend to L-splines of order 4, providing criteria for fast algorithms based on matrix diagonal dominance.

Findings

Fast algorithms for L-spline interpolation are possible under certain matrix conditions.

The matrix R associated with L-splines can be strictly diagonally dominant.

The approach generalizes cubic spline methods to higher-order differential operators.

Abstract

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.