On the saturation class for spline functions

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.