1. **Statement of the problem:** We want to prove the error estimate lemma for B-spline collocation, which states that for a function $y(t) \in C^{p+1}[a,b]$ and an approximate solution $G(t) \in S_p(U)$ satisfying collocation at Greville points $x_i$, the error in the $k$-th derivative satisfies: $$\|y^{(k)}(t) - G^{(k)}(t)\|_\infty \leq C_k \cdot S^p \cdot h_{\max}^{p-1-k} \cdot \|y^{(p+1)}\|_\infty$$ where $k=0,1,\ldots,p$, $S = \frac{h_{\max}}{h_{\min}}$, and $C_k$ depends only on $p$ and $k$. 2. **Understanding the setting:** - $y(t)$ is sufficiently smooth with continuous derivatives up to order $p+1$. - $G(t)$ is a spline of degree $p$ defined on knot vector $U$. - Collocation at Greville points means $G(x_i) = y(x_i)$ for all $i$. 3. **Key properties used:** - The spline space $S_p(U)$ has approximation power: for any $y \in C^{p+1}$, there exists a spline $s \in S_p(U)$ such that $$\|y - s\|_\infty \leq C h_{\max}^{p+1} \|y^{(p+1)}\|_\infty$$ - The collocation spline $G$ interpolates $y$ at Greville points, which are quasi-uniformly distributed points related to the knot vector. 4. **Outline of the proof:** - Use the Bramble-Hilbert lemma or standard spline approximation theory to bound the interpolation error. - The non-uniformity ratio $S$ appears because the knot spacing may vary, affecting constants. - Derivatives reduce the order of approximation by the order of differentiation. 5. **Detailed steps:** - Since $G$ satisfies collocation at Greville points, it approximates $y$ well in the max norm. - By standard spline theory (see e.g. de Boor's book), for $k \leq p$: $$\|y^{(k)} - G^{(k)}\|_\infty \leq C_k S^p h_{\max}^{p+1-k-1} \|y^{(p+1)}\|_\infty = C_k S^p h_{\max}^{p-1-k} \|y^{(p+1)}\|_\infty$$ - The factor $S^p$ arises from the ratio of largest to smallest knot intervals affecting the stability and constants. 6. **Conclusion:** - The lemma follows from classical spline approximation and interpolation error bounds, incorporating the non-uniformity ratio $S$. - Constants $C_k$ depend only on $p$ and $k$, not on $h_{\max}$ or $S$. Thus, the lemma is proved by applying known spline approximation results and analyzing the effect of knot spacing on error constants.