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EfficientL 2 approximation by splines

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Abstract

LetS k N (t) be the linear space ofk-th order splines on [0, 1] having the simple knotst i determined from a fixed functiont by the rulet i=t(i/N). In this paper we introduce sequences of operators {Q N } N =1 fromC k[0, 1] toS k N (t) which are computationally simple and which, asN→∞, give essentially the best possible approximations tof and its firstk−1 derivatives, in the norm ofL 2[0, 1]. Precisely, we show thatN k−1(‖(f−Q N f)i‖−dist2(f (1),S k−1 N (t)))→0 fori=0, 1, ...,k−1. Several numerical examples are given.

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The research of this author was partially supported by the National Science Foundation under Grant MCS-77-02464

The research of this author was partially supported by the U.S. Army Reesearch Office under Grant No. DAHC04-75-G-0816

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Barrow, D.L., Smith, P.W. EfficientL 2 approximation by splines. Numer. Math. 33, 101–114 (1979). https://doi.org/10.1007/BF01396498

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