1. **Problem Statement:** We want to prove the error estimate lemma for B-spline collocation: $$\| y^{(k)} - G^{(k)} \|_{\infty} \leq C_k \cdot S^{p} \cdot h_{max}^{p+1-k} \cdot \| y^{(p+1)} \|_{\infty}$$ where $y \in C^{p+1}[a,b]$, $G$ is the spline approximation, $S = \frac{h_{max}}{h_{min}}$ measures mesh non-uniformity, and $C_k$ is a constant. 2. **Step 1: Approximation Error Bound (Schumaker's Theorem 6.20)** From Schumaker, for any $y(t) \in C^{p+1}[a,b]$, the spline quasi-interpolant $Q_y$ satisfies: $$\| y - Q_y \|_{\infty} \leq D_p h_{max}^{p+1} \| y^{(p+1)} \|_{\infty}$$ This shows the approximation error depends on the mesh size $h_{max}$ and the smoothness of $y$. 3. **Step 2: Stability Constant Bound (Lyche & Mørken's Theorem 9.15)** The stability constant $K_p$ of the spline collocation operator satisfies: $$K_p \leq C_K S^{p}$$ where $S = \frac{h_{max}}{h_{min}}$ measures how non-uniform the mesh is. 4. **Step 3: Combining Approximation and Stability** The collocation spline $G$ satisfies: $$\| y - G \|_{\infty} \leq K_p \| y - Q_y \|_{\infty}$$ Using the bounds from steps 1 and 2: $$\| y - G \|_{\infty} \leq C_K S^{p} \cdot D_p h_{max}^{p+1} \| y^{(p+1)} \|_{\infty}$$ Letting $C = C_K D_p$, we get: $$\| y - G \|_{\infty} \leq C S^{p} h_{max}^{p+1} \| y^{(p+1)} \|_{\infty}$$ 5. **Step 4: Extending to Derivatives (Christara & Ng's Remark 2)** For derivatives of order $k$ ($0 \leq k \leq p$), the error estimate generalizes to: $$\| y^{(k)} - G^{(k)} \|_{\infty} \leq C_k S^{p} h_{max}^{p+1-k} \| y^{(p+1)} \|_{\infty}$$ This accounts for the loss of smoothness order when differentiating. 6. **Summary:** The lemma follows by combining the approximation error bound, the stability constant bound, and the effect of mesh non-uniformity on the collocation spline and its derivatives. This detailed proof shows how the constants and mesh parameters influence the error in spline collocation approximations. \boxed{}