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Monotonicity of the error of optimal segmented approximations

Monotonie des Fehlers optimaler Segmentapproximationen

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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.

Zusammenfassung

Optimale Segmentapproximationen mit freien Knoten für eine stetige Funktion haben (abgesehen von Trivialfällen) streng monoton fallenden Fehler mit wachsender Zahl der Knoten, wenn das Fehlerfunktional streng isoton ist und die Segmente zum linearen Raum der Lösungen einer linearen homogenen Differentialgleichung gehören. Allgemeiner, der Fehler wird kleiner, wenn zwei zusätzliche Knoten zugelassen werden, vorausgesetzt, daß die Segmente zu irgendeiner Menge stetiger Funktionen, die additive Konstanten beinhaltet, gehören.

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References

  1. Kioustelidis, J. B.: Optimal segmented approximations. Computing24, 1–8 (1980).

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  2. de Boor, C.: On the approximation by γ-polynomials. In: Approximation with special emphasis on spline functions (Schoenberg, I. J., ed.), pp. 157–183. New York: Academic Press.

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Kioustelidis, J.B. Monotonicity of the error of optimal segmented approximations. Computing 28, 69–74 (1982). https://doi.org/10.1007/BF02237997

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  • DOI: https://doi.org/10.1007/BF02237997

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