On the saturation class for spline functions
HTML articles powered by AMS MathViewer
- by Franklin Richards PDF
- Proc. Amer. Math. Soc. 33 (1972), 471-476 Request permission
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 $\left \| {f - {\mathcal {S}_k}({\Delta _n})} \right \| + \left \| {f - {\mathcal {S}_k}({\Delta _{n + 1}})} \right \| \geqq c{n^{ - k - 1}}$ unless f is a polynomial of degree at most k. Here we show if $0 < \alpha \leqq 1$, then $\left \| {f - {\mathcal {S}_k}({\Delta _n})} \right \| = O({n^{ - k - \alpha }})$ 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
- Dieter Gaier, Saturation bei Spline-Approximation und Quadratur, Numer. Math. 16 (1970), 129–140 (German). MR 273816, DOI 10.1007/BF02308865 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). F. Richards, Convergence of natural spline interpolants on uniform subdivisions (to appear).
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 33 (1972), 471-476
- MSC: Primary 41A15
- DOI: https://doi.org/10.1090/S0002-9939-1972-0294958-4
- MathSciNet review: 0294958