## 7 Results of Numerical Experiments and Discussion  
We compare the performance of expansions and interpolations using the function  
\[f(x)=\frac{1}{11/10+x}, \tag{180}\]which is non-periodic and infinitely differentiable on \([-1,1]\), with large derivatives near \(x=-1\). Figure 1 displays the maximum absolute errors for the system \(\mathcal{H}_{1}\) on different intervals, plotted on a logarithmic scale. The \"T.F.S.\" stands for the truncated Fourier series \(\mathcal{F}_{N}^{q}[f]\). The value \(m=0\) corresponds to the traditional interpolation based on \(\mathcal{H}_{1}\). The left part of Figure 1 illustrates the errors over the entire interval \([-1,1]\). It confirms our earlier statement that although all expansions and interpolations share the same convergence rate on \([-1,1]\), the QP interpolations are superior due to a smaller constant in the leading term of the asymptotic error. Traditional interpolations are less accurate than the expansions on \([-1,1]\), which is expected since the expansions are optimal in the \(L_{2}[-1,1]\) norm. However, starting from \(m=1\), the QP interpolations outperform both the expansions and traditional interpolations, highlighting one of the advantages of QP interpolations. The right part of Figure 1 focuses on the interval \([-0.8,0.8]\). It demonstrates that interpolations have asymptotically better convergence rates than Fourier expansions on \([-0.8,0.8]\), and the accuracy increases with larger values of the parameter \(m\). Notably, traditional interpolations exhibit a better convergence rate than the expansions on the interval \([-0.8,0.8]\). Figure 2 shows similar results for the expansions and interpolations for the system \(\mathcal{H}_{2}\) corresponding to \(q=2\). The QP interpolations are more accurate away from the endpoints \(x=\pm 1\) (the right part of Figure 2) than the expansions and traditional interpolations. On the entire interval, the expansions are more accurate than the traditional interpolation but less accurate than the QP interpolations. The same behavior is observed for the expansions and interpolations for other systems \(\mathcal{H}_{q}\) (see Figures 3 and 4). We can conclude that by increasing the value of parameter \(q\), we increase the convergence rate by \(O(N)\) away from the endpoints \(x=\pm 1\). On the entire interval, \([-1,1]\), the convergence rate is \(O(N^{-1})\), but the constant at the leading term of the asymptotic error is decreasing while increasing the values of \(q\) and \(m\). In the  
Figure 1: The errors of expansions and interpolations for the system \(\mathcal{H}_{1}\) corresponding to \(q=1\). Figure 2: The errors of expansions and interpolations for the system \(\mathcal{H}_{2}\) corresponding to \(q=2\). following two figures, we show the maximum absolute errors on \([-0.8,0.8]\) and \([-1,1]\) in the logarithmic scale to reveal these dynamics. We used a moderate number of coefficients \(N=64\), and values of \(1\leq q\leq 7\) and \(0\leq m\leq 15\). The value \(m=0\) corresponds to the traditional interpolation. Both figures show that the accuracy of the QP interpolation is proportional to \(q+2m\) which can be observed by similar colors showing identical precisions on the logarithmic scale. This means that the convergence properties of the QP interpolations can be improved by increasing either \(q\) or \(m\). However, calculating the eigenvalues for large \(q\) is connected with more computational difficulties than increasing the value of \(m\) for the fixed value of \(q\). It is possible to tend \(m\) gradually to infinity, as \(\log(N)\). The latter will provide convergence  
Figure 4: The maximum of the absolute errors of interpolations on interval \([-0.8,0.8]\) in the logarithmic scale. The value \(m=0\) corresponds to the traditional interpolation. The values \(m\geq 1\) correspond to the QP interpolations. The value \(q=1\) corresponds to the QP interpolations by the modified Fourier system \(\mathcal{H}_{1}\) with the explicit implementation (62), where \(N=64\).