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 $h = \max x_{j+1} - x_j$ then for $\pi f$ a cubic spline interporlant over the mesh then 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) satisfices

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

They also say (eq 86) that seems pausible 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?

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?

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 $h = \max x_{j+1} - x_j$ then for $\pi f$ a cubic spline interporlant over the mesh then 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) satisfices

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

They also say (eq 86) that seems pausible that the inequality

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

hold even for non uniform meshes.

Has this result been established in the literature for a uniform mesh?

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 $h = \max x_{j+1} - x_j$ then for $\pi f$ a cubic spline interporlant over the mesh then 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) satisfices

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

They also say (eq 86) that seems pausible 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?

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 $h = \max x_{j+1} - x_j$ then for $\pi f$ a cubic spline interporlant over the mesh then 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) satisfices

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

They also say (eq 86) that seems pausible that the inequality

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

Has this result been established in the literature for a uniform mesh?

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 $h = \max x_{j+1} - x_j$ then for $\pi f$ a cubic spline interporlant over the mesh then 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) satisfices

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

They also say (eq 86) that seems pausible that the inequality

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

hold even for non uniform meshes.

Has this result been established in the literature for a uniform mesh?