## 2 Quasi-Periodic Interpolations for the Classical Trigonometric System  
In this section, we delve deeper into QP interpolations that are exact for the classical trigonometric system \(\mathcal{H}_{class}\). Consider the problem of interpolating a function \(f\) on a uniform grid \(\{x_{k}\}\) within the interval \([-1,1]\), where  
\[x_{k}:=\frac{2k}{2N+1},\ |k|\leq N. \tag{10}\]  
To address convergence issues, we extend the interpolation of \(f\) to the interval \([-1/\sigma,1/\sigma]\) using a uniform grid \(\{x_{k}^{*}\}\) in \([-1,1]\) that includes both endpoints:  
\[x_{k}^{*}:=\frac{k}{N},\ |k|\leq N, \tag{11}\]  
and aim for exactness with respect to the system of functions  
\[\{e^{j\pi m\sigma x}\}_{n=-N}^{N}, \tag{12}\]  
where  
\[\sigma_{N,m}:=\sigma:=\frac{2N}{N_{m}}, \tag{13}\]  
and  
\[N_{m}:=2N+2m+1,\ m=1,2,\ldots. \tag{14}\]  
Here, all basis functions are periodic over the interval \(2/\sigma\), which approaches 2 as \(N\) approaches infinity. This is the essence of the term \"quasi-periodic\" in this context. To further clarify, consider the change of variable  
\[t=\sigma x,\ t\in[-1,1],\ x\in[-1/\sigma,1/\sigma] \tag{15}\]which transforms the grid \(\{x_{k}^{*}\}\in[-1,1]\) into \(\{x_{k}^{m}\}\) for \(|k|\leq N\). \[x_{k}^{m}:=\frac{2k}{N_{m}},\ |k|\leq N. \tag{16}\]  
In terms of the variable \(t\), we are solving a classical trigonometric interpolation problem on a non-uniform grid \(\{x_{k}^{m}\}\), \(|k|\leq N\), since some of the \(2m\) points are missing from the uniform grid \(\{x_{k}^{m}\}\), \(|k|\leq N+m\). Consequently, the QP interpolation requires solving a system of linear equations with \(2m\) unknowns. For the classical trigonometric system, this system's matrix is Vandermonde, allowing for an explicit solution of the unknowns, as shown in [66]. Denote the QP interpolation for the system \(\mathcal{H}_{class}\) by \(I_{N,m}^{0}[f](x)\). It can be expressed as  
\[I_{N,m}^{0}[f](x):=\sum_{k=-N}^{N}f\bigg{(}\frac{k}{N}\bigg{)}c_{k}(x), \tag{17}\]  
where  
\[c_{k}(x):=\frac{1}{N_{m}}\Bigg{(}\sum_{n=-N}^{N}\omega_{2n}^{-k}e^{i\pi n\sigma x }-\sum_{\ell=1}^{2m}\omega_{2\ell+2N}^{N+2m-k}\sum_{j=1}^{2m}v_{\ell,j}^{-1} \sum_{n=-N}^{N}\omega_{2n}^{j+N}e^{i\pi n\sigma x}\Bigg{)}, \tag{18}\]  
and  
\[\omega_{n}(N,m):=\omega_{n}:=e^{\frac{i\pi n}{N_{m}}}. \tag{19}\]  
Here, \(v_{\ell,j}^{-1}\) are elements of the inverse of the Vandermonde matrix:  
\[x_{k}^{*}:=\frac{k}{N},\ |k|\leq N, \tag{11}\]0  
Efficient algorithms for computing inverses of Vandermonde matrices can be found in [78; 79; 80; 81]. From here, the elements \(v_{\ell,j}^{-1}\) have the following explicit form  
\[v_{\ell,j}^{-1}=-\frac{1}{\omega_{2\ell+2N}^{j}\prod_{ \begin{subarray}{c}k= 1\\ k\
eq\ell\end{subarray} }^{2m}(\omega_{2\ell+2N}-\omega_{2k+2N})}\sum_{t=0}^{j -1}\gamma_{t}\omega_{2\ell+2N}^{t},\ \ \ell,j=1,\ldots,2m, \tag{21}\]  
where \(\gamma_{j}\) are the coefficients of the following polynomial  
\[\prod_{k=1}^{2m}(x-\omega_{2k+2N})=\sum_{t=0}^{2m}\gamma_{t}x^{t}. \tag{22}\]  
The QP interpolation \(I_{N,m}^{0}[f](x)\) can also be expressed in terms of discrete Fourier coefficients:  
\[I_{N,m}^{0}[f](x)=\sum_{n=-N}^{N}\Bigg{(}\tilde{f}_{n,m}^{0}-\sum_{\ell=1}^{2 m}\theta_{n,\ell}\tilde{f}_{\ell+N,m}^{0}\Bigg{)}e^{i\pi n\sigma x}, \tag{23}\]  
where  
\[\tilde{f}_{n,m}^{0}:=\frac{1}{N_{m}}\sum_{k=-N}^{N}f\bigg{(}\frac{k}{N}\bigg{)} \omega_{k}^{-2n}, \tag{24}\]  
and  
\[\theta_{n,\ell}:=\omega_{2\ell+2N}^{-N-1}\sum_{j=1}^{2m}v_{\ell,j}^{-1}\omega _{2n}^{j+N}. \tag{25}\]  
The convergence of the QP interpolations has been studied in [66; 67], with the following results:1. The pointwise convergence rate of \(I^{0}_{N,m}[f](x)\) for \(m\geq 1\) is \(O(N^{-2m-1})\) as \(N\to\infty\), away from the endpoints \(|x|<1\). In contrast, the classical interpolation has a rate of \(O(N^{-1})\). Thus, the improvement is of order \(O(N^{2m})\). 2. The convergence in the \(L_{2}(-1,1)\) norm is \(O(N^{-1/2})\) for both QP and classical interpolations, but the QP interpolation is significantly more accurate. 3.