On Construction of Fourth Order Chebyshev Splines

It is an important fact that general families of Chebyshev and L-splines can be locally represented, i.e. there exists a basis of B-splines which spans the entire space. We develop a special technique to calculate with 4 order Chebyshev splines of minimum deficiency on nonuniform meshes, which leads to a numerically stable algorithm, at least in case one special Hermite interpolant can be constructed by stable explicit formulæ. The algebraic derivation of the algorithm involved makes it possible to apply the construction to L-splines. The underlying idea is an Oslo type algorithm, combined with the known derivative formula for Chebyshev splines. We then show that weighted polynomial and tension spline spaces satisfy the conditions imposed, and show how to apply the above general techniques to obtain local representations.