## 4 Error Analysis
### Practical Guidelines for Parameter Selection  
The asymptotic analysis reveals dependencies on the parameters \(\lambda\) (for interpolation) and \(\lambda_{q}\) (for quadrature), which play crucial roles in determining the method's accuracy and efficiency. Here, we provide practical guidance for selecting these parameters to balance interpolation error, quadrature error, and computational cost. The effectiveness of the GBFA method for approximating the RLFI hinges on the appropriate selection of parameters \(\lambda\) and \(\lambda_{q}\). Building upon established numerical approximation principles, our analysis incorporates specific considerations for RLFI computation. In particular, we identify the following effective operational range for the SG parameters \(\lambda\) and \(\lambda_{q}\):  
\[\mathcal{T}_{\epsilon,r}=\bigg{\{}\gamma\mid-\frac{1}{2}+\epsilon\leq\gamma \leq r,\,0<\epsilon\ll 1,\,r\in\mathbf{\Omega}_{1,2}\bigg{\}}. \tag{49}\]  
This range, previously recommended by [PERSON] and [PERSON] [PERSON] (2018) based on extensive theoretical and numerical testing consistent with broader spectral approximation theory, helps avoid numerical instability caused by increased extrapolation effects associated with larger positive SG indices. Furthermore, SG polynomials exhibit blow-up behavior as their indices approach \(-0.5^{+}\). Within \(\mathcal{T}_{\epsilon,r}\), we observe a crucial balance between theoretical convergence and numerical stability for RLFI computation, a finding corroborated by our numerical investigations using the GBFA method, which consistently demonstrate superior performance across various test functions with parameter choices in this range. Based on this analysis of error bounds and numerical performance, we recommend the following parameter selection strategies for RLFI approximation. * \(\forall_{\text{r}}\,n\) and \(n_{q}\), the selection of \(\lambda\in\mathcal{T}_{c,r}\) is feasible. Furthermore, choosing smaller \(\lambda_{q}\in\mathcal{T}_{c,r}\) generally improves quadrature accuracy, with the notable exception of \(\lambda_{q}=0.5\), where the quadrature error is often minimized, as we demonstrate later in Figures 4 and 5. * \(\forall_{\text{r}}\,n\) and \(n_{q}\): **- For precision computations:** Select \(\lambda\in\boldsymbol{\Omega}_{-0.5+\varepsilon,0}\setminus N_{\delta}( \lambda^{*})\) and \(\lambda_{q}\in\mathcal{T}_{c,r}\), where \[\lambda_{q}< \begin{cases}\lambda,&\text{if }n\sim n_{q},\\ 3/2,&\text{if }n\gg n_{q},\end{cases} \] (50) with \(\lambda^{*}\) defined by Equation (33). Here, \(\boldsymbol{\Omega}_{-0.5+\varepsilon,0}\) is a subset of the recommended interval \(T_{c,r}\) for SG indices. Positive values of \(\lambda\) are excluded from \(\mathcal{T}_{c,r}\), as the polynomial error factor \(n^{-\frac{1}{2}-n+\lambda}\) increases with positive \(\lambda\), as shown in Equation (27). The \(\delta\)-neighborhood \(N_{\delta}(\lambda^{*})\) is excluded because \(\theta_{\alpha,\lambda}\), the leading factor in the asymptotic interpolation error bound, peaks at \(\lambda=\lambda^{*}\), potentially increasing the error. By avoiding this neighborhood, parameter choices that could amplify interpolation errors are circumvented, ensuring robust performance for high-precision applications. * **For standard computational scenarios:** Utilize \(\lambda=\lambda_{q}=0\), which corresponds to shifted Chebyshev approximation. This recommendation employs the well-established optimality of Chebyshev approximation for smooth functions, offering a robust default that balances accuracy and efficiency for RLFI approximation. Figure 4: Logarithmic absolute errors of the RLFI approximations for the power function \(f\), computed using the GBFA method. The fractional order is set to \(\alpha=0.5\), and approximations are evaluated at \(t=0.5\). Gegenbauer interpolant degrees match the function’s degrees for \(n=3:2:11\). The figure presents errors under different conditions: (**Left**): Varying \(\lambda\) with fixed \(\lambda_{q}=0.5\) and \(n_{q}=4\).