Unicity of Best Mean Approximation by Second Order Splines with Variable Knots

Abstract

Let $S_N^2$ denote the nonlinear manifold of second order splines defined on [0, 1] having at most $N$ interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function $f$ by elements of $S_N^2$. Approximation relative to the ${L_2}[0,1]$ norm is treated first, with the results then extended to the best ${L_1}$ and best one-sided ${L_1}$ approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function $f$ satisfying $f” > 0$ has a unique best approximant from $S_N^2$ provided either $\log f”$ is concave, or $N$ is sufficiently large, $N \geqslant {N_0}(f)$; for any $N$, there is a smooth function $f$, with $f” > 0$, having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.