2
$\begingroup$

Hall & Meyer, 1976, J. Approx. Theory, show for $f \in C^4[a,b]$ and a mesh $a = x_1, \ldots x_n = b$ with $h = \max x_{j+1} - x_j$, for $\pi f$ a cubic spline interpolant over the mesh for some constants $C_k$

$$ \| (f - \pi f)^{(r)} \|_\infty \leq C_r \|f^{(4)}\|_{\infty}h^{4-r} ,\quad 0 \leq r \leq 3.$$

In the same work, Theorem 6 states that for $f\in C^{2m}[a,b]$ the type II $2m-1$ degree spline interpolant (in a uniform mesh) satisfies

$$ \| f -\pi f \|_\infty \leq C_m \|f^{(2m)}\|_\infty h^{2m}.$$

They also say (eq. 86) that it seems plausible that the following inequality may hold even for non uniform meshes,

$$ \| (f -\pi f)^{(r)} \|_\infty \leq C_{m,r} \|f^{(2m)}\|_\infty h^{2m-r} \quad 0 \leq r \leq m.$$

Has this result been established in the literature? Does it hold at least for a uniform mesh?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ you can prove exactness for polynomials and then it follows see for instance Thm 3.5.4 (page 73) here: research-collection.ethz.ch/bitstream/handle/20.500.11850/… $\endgroup$
    – user100927
    Commented Oct 2, 2019 at 14:03
  • $\begingroup$ @user100927 Thanks! In Thm 3.5.4 I am having trouble to understand the identity used for $\frac{d^l}{\partial x^l}I_R(u-p)(x)$. But i assume it would be easier in my setting. If $p$ is the taylor expansion in $x$ of $f$ what would $(\pi(f - p))^r$ be?. I understand $f-p$ is an $2m$ times diferentiable in $x$ such that $| f(y)-p (y)| \leq \|f^{(2m)}\|_\infty(x-y)^{(2m)}$. $\endgroup$
    – Manuel
    Commented Oct 2, 2019 at 15:23

0

Reset to default

You must log in to answer this question.