Abstract
Optimal segmented approximations with free knots for a continuous function have (with the exception of trivial cases) strictly monotonically decreasing error with increasing number of knots, if the error functional is strictly isotonic and the approximation segments belong to the linear space of solutions of a linear homogeneous differential equation. More generally, the error is certain to decrease, if two additional knots are allowed, provided that the segments belong to any set of continuous functions, which includes additive constants.
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