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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the saturation class for spline functions

Author: Franklin Richards
Journal: Proc. Amer. Math. Soc. 33 (1972), 471-476
MSC: Primary 41A15
MathSciNet review: 0294958
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Abstract: Let $ {\mathcal{S}_k}({\Delta _n})$ be the space of piecewise polynomials of degree at most k on [0, 1] possessing jumps at $ 1/n,2/n, \cdots ,n - 1/n$. Recently Gaier has shown that under the supremum norm $ \left\Vert {f - {\mathcal{S}_k}({\Delta _n})} \right\Vert + \left\Vert {f - {\mathcal{S}_k}({\Delta _{n + 1}})} \right\Vert \geqq c{n^{ - k - 1}}$ unless f is a polynomial of degree at most k. Here we show if $ 0 < \alpha \leqq 1$, then $ \left\Vert {f - {\mathcal{S}_k}({\Delta _n})} \right\Vert = O({n^{ - k - \alpha }})$ if and oniy if $ f \in {C^k}[0,1]$ and $ {f^{(k)}}$ satisfies a Lipschitz condition of order $ \alpha $. In addition, a result similar to Gaier's is given.

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

  • [1] Dieter Gaier, Saturation bei Spline-Approximation und Quadratur, Numer. Math. 16 (1970), 129–140 (German). MR 0273816,
  • [2] V. A. Popov and Bl. Kh. Sendov, Classes characterized by best-possible approximation by spline functions, Math. Notes 8 (1970), no. 2, 550-557 (translated from Mat. Zametki).
  • [3] F. Richards, Convergence of natural spline interpolants on uniform subdivisions (to appear).

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Keywords: Spline function, saturation class
Article copyright: © Copyright 1972 American Mathematical Society

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