Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 (42 #8692)
  • [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).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 41A15

Retrieve articles in all journals with MSC: 41A15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0294958-4
PII: S 0002-9939(1972)0294958-4
Keywords: Spline function, saturation class
Article copyright: © Copyright 1972 American Mathematical Society