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

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