Interpolation by Spline Functions

Abstract

Even an elementary study of the interpolation problem

$$s\left( {{t_i}} \right) = {f_i},\quad i = 0,...,n + 1 $$

in the simplest linear case, i.e., when s ∈ S ₁(x ₁,... ,x _n ), shows that the solvability of the corresponding system depends entirely on the mutual location of the interpolation nodes $t = \left\{ {{t_i}} \right\}_0^{n + 1}$ and the spline knots $x = \left\{ {{x_i}} \right\}_i^n$ For example, in the case t _i = x _i , i = 1,... ,n, the problem has a unique solution: the piecewise linear function with vertices at (t _i, f _i ), i = 0,..., n + 1 On the other hand, in the case where three or more interpolation nodes are situated between two consecutive x _i ’s, the problem becomes unresolvable. We shall give here a complete characterization of the Hermite interpolation problem by spline functions with multiple knots. The B-spline representation of s leads us to the study of the corresponding collocation matrix {B _i (t _j)}.