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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Barrow, D.L., Smith, P.W.: Asymptotic properties of bestL 2[0,11] approximation by splines with variable knots, Quart. of Appl. Math.,36, 293–304 (1978)
de Boor, C.: On calculating withB-splines, J. Approx. Theory,6, 50–62 (1972)
de Boor, C.: The quasi-interpolant as a tool in elementary spline theory. In: Approximation Theory pp. 269–276 (G.G. Lorentz, ed.). New York: Academic Press 1973
de Boor, C.: On local linear functionals which vanish at allB-splines but one, In: Theory of Approximation with Applications pp. 120–145 (A. Law and B. Sahney, eds.). New York: Academic Press 1976
Lyche, T., Schumaker, L.L.: Local spline approximation methods. J. Approx. Theory,15, 294–375 (1975)
Schoenberg, I.J.: Monosplines and quadrature formulae, in Theory and Applications of Spline Functions pp. 157–207 (T.N.E. Greville, ed.), New York: Academic Press 1969
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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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DOI: https://doi.org/10.1007/BF01396498