## 4 Error Analysis
When \(j\sim n_{q}\), the dominant term in \(\sup\left|\mathfrak{T}_{j,n_{q}}^{\lambda_{q}}(\gamma)\right|\) becomes  
\[\frac{1}{(n+1)!\Gamma(\alpha+1)\bar{K}_{n+1}^{\lambda}}=\frac{2^{-2n-1}(2 \lambda+1)\Gamma(\lambda+2)\Gamma(n+2\lambda+1)}{(\lambda+1)\Gamma(\alpha+1) \Gamma(2\lambda+2)\Gamma(n+2)\Gamma(n+\lambda+1)}, \tag{29}\]7  
Exponential decay occurs when  
\[\frac{1}{(n+1)!\Gamma(\alpha+1)\bar{K}_{n+1}^{\lambda}}=\frac{2^{-2n-1}(2 \lambda+1)\Gamma(\lambda+2)\Gamma(n+2\lambda+1)}{(\lambda+1)\Gamma(\alpha+1) \Gamma(2\lambda+2)\Gamma(n+2)\Gamma(n+\lambda+1)}, \tag{29}\]8  
On the other hand, the dominant term in the error bound \(\forall j\gg n_{q}\) is given by  
\[\frac{1}{(n+1)!\Gamma(\alpha+1)\bar{K}_{n+1}^{\lambda}}=\frac{2^{-2n-1}(2 \lambda+1)\Gamma(\lambda+2)\Gamma(n+2\lambda+1)}{(\lambda+1)\Gamma(\alpha+1) \Gamma(2\lambda+2)\Gamma(n+2)\Gamma(n+\lambda+1)}, \tag{29}\]9  
For convergence, we require  
\[T_{2}(\lambda)=\left[\frac{(1+\lambda)\operatorname{csch}\!\left(\tfrac{1}{1+2 \lambda}\right)}{1+2\lambda}\right]^{\frac{1}{2}+\lambda}. \tag{30}\]0  
Given \(j\gg n_{q},j^{2}/n_{q}\to\infty\), and \(\frac{etj^{2}\eta^{\frac{1}{2}-1}}{2\alpha n_{q}}\) typically diverges unless  
\[T_{2}(\lambda)=\left[\frac{(1+\lambda)\operatorname{csch}\!\left(\tfrac{1}{1+2 \lambda}\right)}{1+2\lambda}\right]^{\frac{1}{2}+\lambda}. \tag{30}\]1The relative choice of \(\lambda\) and \(\lambda_{q}\) in either case controls the error bound's decay rate. In particular, choosing \(\lambda_{q}<\lambda\) ensures faster convergence rates when \(j\sim n_{q}\) due to presence of the polynomial factor \(n_{q}^{\lambda_{q}-\lambda}\). For \(j\gg n_{q}\), choosing \(\lambda_{q}<3/2\) accelerates the convergence if Condition (43) holds. The following theorem provides a rigorous asymptotic bound on the total truncation error for the RLFI approximation, combining both sources of error, namely, the interpolation and quadrature errors. **Theorem 5** (Asymptotic Total Truncation Error Bound). : _Suppose that \(f\in C^{n+1}(\mathbf{\Omega}_{1})\) is approximated by the GBFA interpolant (7), and the assumptions of Theorems 1 and 3 hold true. Then the total truncation error in the RLFI approximation of \(f\), denoted by \(\ast^{\lambda}\varepsilon_{n,n_{q}}^{\lambda,\lambda_{q}}(t,\xi,\eta)\), arising from both the series truncation (7) and the quadrature approximation (21), is asymptotically bounded above by_  
\[ \begin{array}{l}\left|\ast\varepsilon_{n,n_{q}}^{\lambda,\lambda_{q}}(t,\xi, \eta)\right|\lesssim\mathcal{A}_{n+1}\,\theta_{\alpha,\lambda}\left(\frac{ \xi}{\xi}\right)^{n}n^{-\frac{3}{2}-n+\lambda}\,\mathrm{Y}_{\sigma^{\lambda}} (n)+\frac{\mathcal{A}_{0}\,\omega^{\text{upper}}\mu}{\lambda_{\max}^{\lambda }\Gamma(\alpha+1)}\,n(n-n_{q})\eta^{\frac{n_{q}(1-\alpha)}{\alpha}}{}_{2} \mathrm{Y}_{D^{\lambda},\rho^{\lambda_{q}}}(n,n_{q})\,\Im_{n\geq n_{q}+1}\\ \left\{\begin{array}{l}\mu_{\lambda,\lambda_{q}}n_{q}^{\lambda_{q}-\lambda} \left(\frac{2t}{\alpha}\right)^{n_{q}},\quad n\sim n_{q},\\ \frac{\gamma_{q}^{\lambda_{q}}n_{q}^{\lambda_{q}/2-\lambda_{q}}}{\left(\frac{ \mu t}{2n_{q}}\right)}^{n_{q}},\quad n\gg n_{q},\end{array}\right.\end{array}  \tag{44}\]  
\[\forall_{tl}n,n_{q}\text{, where} \tag{45}\]  
\[\frac{1}{\lambda_{\max}^{\lambda}}=\left\{ \begin{array}{l}\frac{1}{\lambda _{n}^{\lambda}},\quad\lambda\in\mathbb{R}_{0}^{+}\,,\\ \frac{1}{\lambda_{\max}^{\lambda}},\quad\lambda\in\mathbb{R}_{-1/2}^{-1}, \end{array} \right. \tag{46}\]  
\[{}_{2}\mathrm{Y}_{D^{\lambda},\rho^{\lambda_{q}}}(n,n_{q})= \begin{cases}1,& \lambda\in\mathbb{R}_{0}^{+}\,,\quad\lambda_{q}\in\mathbb{R}_{0}^{+}\,,\\ D^{\lambda}n^{-\lambda},&\lambda\in\mathbb{R}_{-1/2}^{-1},\quad\lambda_{q}\in \mathbb{R}_{0}^{+}\,,\\ p^{\lambda_{q}}n_{q}^{-\lambda_{q}},&\lambda\in\mathbb{R}_{0}^{+}\,,\quad \lambda_{q}\in\mathbb{R}_{-1/2}^{-},\\ D^{\lambda}\rho^{\lambda_{q}}n^{-\lambda_{q}}n_{q}^{-\lambda_{q}},&\lambda \in\mathbb{R}_{-1/2}^{-1},\end{cases}