## 4 Error Analysis
Theorem 2 manifests that the error bound is influenced by the smoothness of the function \(f\) (through its derivatives) and the specific values of \(\alpha\) and \(\lambda\) with super-exponentially decay rate as \(n\to\infty\), which guarantees that the error becomes negligible even for moderate values of \(n\). By looking at the bounding constant \(\theta_{\alpha,\lambda}\), we notice that, while holding \(\lambda\) fixed, the factor \(\alpha^{-\frac{1}{2}-\alpha}\) decays exponentially as \(\alpha\) increases because \(\alpha^{-\alpha}\) dominates. For large \(\alpha\), \(\sinh(1/\alpha)\approx 1/\alpha\), so the factor \(\left[\alpha\sinh(1/\alpha)\right]^{-\alpha/2}\) behaves like \(1\). Thus, it does not significantly affect the behavior as \(\alpha\) increases. The factor \(\epsilon^{\alpha+1}\) grows exponentially as \(\alpha\) increases. Combining these observations, the dominant behavior as \(\alpha\) increases is determined by the exponential growth of \(e^{\alpha+1}\) and the exponential decay of \(\alpha^{-\frac{1}{2}-\alpha}\). The exponential decay dominates, so \(\theta_{\alpha,\lambda}\) decays as \(\alpha\) increases, leading to a tighter error bound and improved convergence rate. On the other hand, considering large \(\lambda\) values while holding \(\alpha\) fixed, the factor \((1+2\lambda)^{-\frac{1}{2}-2\lambda}\) decays as \(\lambda\) increases. The factor \(1+1/\big{(}1620(1+\lambda)^{5}\big{)}\) approaches \(1\) as \(\lambda\) increases. For large \(\lambda\), \(\operatorname{csch}\!\left[1/(1+2\lambda)\right]\approx 1+2\lambda\), so the factor \(\left[\frac{(1+\lambda)\operatorname{csch}\!\left(\tfrac{1}{1+2\lambda} \right)}{1+2\lambda}\right]^{\frac{1}{2}+\lambda}\) behaves like \((1+\lambda)^{\frac{1}{2}+\lambda}\). This grows as \(\lambda\) increases. Finally, the factor \(\left[(1+\lambda)\sinh\!\left(\tfrac{1}{1+\lambda}\right)\right]^{\frac{1+ \lambda}{2}}\) behaves like \(1\) as \(\lambda\) increases. Thus, it does not significantly affect the behavior as \(\lambda\) increases. Combining these observations, the dominant behavior as \(\lambda\) increases is determined by the growth of \((1+\lambda)^{\frac{1}{2}+\lambda}\) and the decay of \((1+2\lambda)^{-\frac{1}{2}-2\lambda}\). The decay of \((1+2\lambda)^{-\frac{1}{2}-2\lambda}\) dominates, so \(\theta_{\alpha,\lambda}\) also decays as \(\lambda\) increases. It is noteworthy that \(\theta_{\alpha,\lambda}\) remains finite as \(\lambda\to-1/2\), even though \(\lim_{\lambda\to-0.5^{+}}T_{1}(\lambda)=\infty\), where \(T_{1}(\lambda)=(1+2\lambda)^{-\frac{1}{2}-2\lambda}\). This divergence is offset by the behavior of \(T_{2}(\lambda)\):  
\[T_{2}(\lambda)=\left[\frac{(1+\lambda)\operatorname{csch}\!\left(\tfrac{1}{1+2 \lambda}\right)}{1+2\lambda}\right]^{\frac{1}{2}+\lambda}. \tag{30}\]  
Specifically, for \(\lambda\to-0.5^{+}\), using the approximation \(\operatorname{csch}\!\left(x\right)\approx 2e^{-x}\) for large \(x\):  
\[T_{2}(\lambda)\approx\left[\frac{(1+\lambda)\cdot 2e^{-\frac{1}{1+2\lambda}}}{1+2 \lambda}\right]^{\frac{1}{2}+\lambda}\sim\left[\frac{0.5\cdot 2e^{-\frac{1}{0^{+}}}}{0^{+}} \right]^{0^{+}}\to 0. \tag{31}\]  
Consequently,  
\[\lim_{\lambda\to-0.5^{+}}T_{1}(\lambda)T_{2}(\lambda)=0. \tag{32}\]Hence, \(\vartheta_{a,\lambda}\to 0\) as \(\lambda\to-1/2^{+}\), driven by the product \(T_{1}(\lambda)T_{2}(\lambda)\to 0\). For \(\lambda\in\mathbb{R}_{-1/2^{+}}^{-}\), we notice that \(T_{1}\) is strictly concave with a maximum value at  
\[\lambda^{*}=\frac{-e+e^{\mathrm{W}(e/2)}}{2e}\approx-0.1351, \tag{33}\]  
rounded to 4 significant digits; cf. Theorem A1. Figure 2 shows further the plots of \(T_{2}(\lambda)\) and  
\[T_{3}(\lambda)=\left[(1+\lambda)\sinh\left(\frac{1}{1+\lambda}\right)\right]^ {\frac{1+\lambda}{2}},\]  
where \(T_{2}\) grows at a slow, quasi-linearly rates of change, while \(T_{3}\) decays at a comparable rate; thus, their product remains positive with a small bounded variation.