Abstract
We give a necessary alternation condition for unique local best approximation from Sm,k, the set of splines of degree m with k free knots. This result is related to a conjecture of L.L. Schumaker. Moreover, we give a characterization of functions from the interior of the strong unicity set for S1m,k, the set of splines of degree m with k free simple knots, and show that this set is dense in the unicity set. Then we give a general characterization of suns for strong unicity and show that S1m,k is a set of this type, although it is not a sun.
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