Abstract
We derive a necessary alternation condition for unique local best approximations fromS m, k , the set of splines of degreem withk free knots. This result is related to a conjecture of L. L. Schumaker. Moreover, we give a complete description of functions from the interior of the strong unicity set forS 1 m,k the set of splines of degreem withk free simple knots, and show that this set is dense in the unicity set forS 1 m,k . Finally, we prove a general characterization of suns for strong unicity and show thatS 1 m,k is a set of this type, although, it is not a sun.
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Communicated by Larry L. Schumaker.
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Nürnberger, G. Strong unicity in nonlinear approximation and free knot splines. Constr. Approx 10, 285–299 (1994). https://doi.org/10.1007/BF01263068
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DOI: https://doi.org/10.1007/BF01263068