STRONG UNICITY IN NONLINEAR APPROXIMATION AND FREE KNOT SPLINES

We derive a necessary alternation condition for unique local best appr oximations from S(m,k), the set of splines of degree m with k free kno ts. This result is related to a conjecture of L. L. Schumaker. Moreove r, we give a complete description of functions from the interior of th e strong unicity set for S(m,k)1, the set of splines of degree m with k free simple knots, and show that this set is dense in the unicity se t for S(m,k)1. Finally, we prove a general characterization of suns fo r strong unicity and show that S(m,n)1 is a set of this type, although it is not a sun.