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Another extremal property of perfect splines


Authors: T. N. T. Goodman and S. L. Lee
Journal: Proc. Amer. Math. Soc. 70 (1978), 129-135
MSC: Primary 41A15
DOI: https://doi.org/10.1090/S0002-9939-1978-0481760-9
MathSciNet review: 0481760
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Abstract: Let $ {\mathbf{t}} = \{ {t_i}\} ,i = 1,2, \ldots ,n + k$, be a given sequence in [a, b], $ {f_0} \in W_\infty ^k[a,b],A > {\left\Vert {f_0^{(k)}} \right\Vert _\infty }$, and $ F = \{ f \in W_\infty ^k[a,b]:f\vert{\mathbf{t}} = {f_0}\vert{\mathbf{t}}$ and $ {\left\Vert {{f^{(k)}}} \right\Vert _\infty } \leqslant A\} $. We show that F contains precisely two perfect splines g, h of degree k with $ \vert{g^{(k)}}\vert = \vert{h^{(k)}}\vert = A$ and n interior nodes, and for all $ f \in F$, $ \min (g(x),h(x)) \leqslant f(x) \leqslant \max (g(x),h(x));\forall x \in [a,b]$.


References [Enhancements On Off] (What's this?)

  • [1] Carl De Boor, A remark on perfect splines, Bull. Amer. Math. Soc. 80 (1974), 724-727. MR 0338618 (49:3382)
  • [2] S. Karlin, Some variational problems on certain Sobolev spaces and perfect splines, Bull. Amer. Math. Soc. 79 (1973), 124-128. MR 0308769 (46:7883)
  • [3] G. Glaeser, Prolongement extremal de fonctions differentiables d'une variable, J. Approximation Theory 8 (1973), 249-261. MR 0348068 (50:566)
  • [4] T. N. T. Goodman and S. L. Lee, Some extremal problems involving perfect splines, Comment. Math. (to appear). MR 1887504 (2003a:41006)
  • [5] R. Louboutin, Sur une ``bonne'' partition de l'unité, Le Prolongateur de Whitney, G. Glaeser (ed.), Vol. II, 1967.
  • [6] D. E. McClure, Perfect spline solutions of $ {L_\infty }$ extremal problems by control methods, J. Approximation Theory 15 (1975), 226-242. MR 0397250 (53:1109)
  • [7] I. J. Schoenberg, The perfect B-splines and a time-optimal control problem, Israel J. Math. 10 (1971), 261-275. MR 0320594 (47:9130)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0481760-9
Keywords: Perfect splines
Article copyright: © Copyright 1978 American Mathematical Society

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