1. **Problem Statement:** You want to verify if the step involving inequalities (5.3) to (5.7) correctly proves the lemma about the error bound for derivatives of the function approximation using B-splines, and how to continue from there. 2. **Understanding the given inequalities:** - Inequality (5.3) states a general error bound for the $r$-th derivative of the difference between $f$ and its approximation $Qf$. - Substituting $m=p$, $\sigma=p+1$, $r=k$, $\Delta=h_{max}$, $f=y(t)$, and $Qf=G(t)$ leads to (5.4). - Simplifying the modulus of continuity term $\omega_0$ using the assumption $\omega_0(D^p y(t); h_{max})_\infty \leq C_\omega h_{max} \|y^{(p+1)}\|_\infty$ with $C_\omega \leq 1$ leads to (5.6) and then (5.7). 3. **Is this step correct?** Yes, this is a standard approach in approximation theory: - You start from a general error bound involving moduli of continuity. - You use smoothness assumptions on $y(t)$ (existence and boundedness of derivatives up to order $p+1$). - You replace the modulus of continuity by a bound involving the next derivative and $h_{max}$. - This yields a final error bound of the form $$\|y^{(k)} - G^{(k)}\|_\infty \leq C_1 h_{max}^{p-k+1} \|y^{(p+1)}\|_\infty,$$ which is a classical result for spline approximation error in derivatives. 4. **How to continue the proof?** - Use this error bound to establish convergence rates of the spline approximation and its derivatives. - Show that the constant $C_1$ depends only on $p$ (the spline degree) and not on $h_{max}$. - Possibly use stability results of B-splines (as mentioned) to control norms and ensure well-posedness. - If proving a lemma, explicitly state assumptions on $y(t)$ (smoothness) and on the mesh size $h_{max}$. - You may also want to verify or cite the stability of the B-spline basis to ensure the approximation operator $Q$ behaves well. 5. **Summary:** The step is correct and standard in spline theory. To complete the lemma proof, combine this error bound with stability and smoothness assumptions to conclude the desired approximation properties.