## 3 Quasi-Periodic Interpolations for the Modified Trigonometric System
 \tag{52}\]  
Now, subtracting the equations of (48) from the corresponding equations of (46), we will obtain the following system of equations for unknowns \(\epsilon_{n}^{*}(x)\) having only odd indices  
\[\sum_{n=N}^{N+m-1}\omega_{2n+1}^{j}\epsilon_{2n+1}^{*}(x)+\sum_{n =-N-m}^{-N-1}\omega_{2n+1}^{j}\epsilon_{2n+1}^{*}(x)\\ =\sum_{n=-N}^{N-1}\frac{\omega_{2n+1}^{j}}{2}\left[-e^{\frac{in(2n +1)vx}{2}}+e^{-\frac{in(2n+1)vx}{2}}\right],\ j=N+1,\ldots,N+2m. \tag{53}\]  
Based on the relations (as \(\omega_{2n-2N_{m}+1}=\omega_{2n+1}\))  
\[\sum_{n=-N-m}^{-N-1}\omega_{2n+1}^{j}\epsilon_{2n+1}^{*}(x)=\sum_{n=N+m+1}^{N +2m}\omega_{2n+1}^{j}\epsilon_{2n-2N_{m}+1}^{*}(x),\ j=N+1,\ldots,N+2m, \tag{54}\]  
we obtain the following linear system of equations  
\[\sum_{\ell=1}^{2m}V_{j,\ell}b_{2\ell+2N+1}(x)=\sum_{n=-N}^{N-1}\frac{\omega_{ 2n+1}^{j+N}}{2}\left[-e^{\frac{in(2n+1)vx}{2}}+e^{-\frac{in(2n+1)vx}{2}}\right],\quad j=1,\ldots,2m \tag{55}\]  
with Vandermonde matrix  
\[V_{j,\ell}=\beta_{\ell}^{j-1},\quad j=1,\ldots,2m,\ \ell=1,\ldots,2m, \tag{56}\]  
where  
\[\beta_{\ell}(N,m):=\beta_{\ell}:= \begin{cases}\omega_{2\ell+2N-1},&\ell=1, \ldots,m,\\ \\ \omega_{2\ell+2N+1},&\ell=m+1,\ldots,2m,\end{cases}  \tag{57}\]  
and unknowns  
\[b_{2n+1}(x):= \begin{cases}\omega_{2n-1}^{N+1}\epsilon_{2n-1}^{*}(x),&n=N+1, \ldots,N+m,\\ \\ \omega_{2n+1}^{N+1}\epsilon_{2n-2N_{m}+1}^{*}(x),&n=N+m+1,\ldots,N+2m.\end{cases}  \tag{58}\]  
Similar to (20), the inverse of \(\left[V_{j,\ell}\right]_{j,\ell=1}^{2m}\) can be calculated explicitly. The corresponding elements of the inverse can be written as follows:  
\[V_{\ell,j}^{-1}=-\frac{1}{\beta_{\ell}^{j}\prod_{k=1,k\
eq l}^{2m}(\beta_{ \ell}-\beta_{k})}\sum_{t=0}^{j-1}\delta_{t}\beta_{\ell}^{t},\quad\ell,j=1, \ldots,2m, \tag{59}\]  
where  
\[\prod_{k=1}^{2m}(x-\beta_{k})=\sum_{t=0}^{2m}\delta_{t}x^{t}. \tag{60}\]Now, taking into account (44) and (41), we determine the coefficients \(a_{k}\):  
\[a_{k}(x)=\frac{1}{N_{m}}\Bigg{(}\sum_{n=-2N}^{2N}\frac{\omega_{n}^{ k}}{2}\Big{[}(-1)^{n}e^{\frac{imvx}{2}}+e^{-\frac{imvx}{2}}\Big{]}\\ -\sum_{n=2N+1}^{Nm-1}\omega_{n}^{k}e_{n}^{*}(x)-\sum_{n=-N_{m}+1}^ {-2N-1}\omega_{n}^{k}e_{n}^{*}(x)\Bigg{)},\quad|k|\leq N. \tag{61}\]  
Then, the QP interpolation for the system \(\mathcal{H}_{1}\) has the following explicit form:  
\[\mathcal{I}_{N,m}^{1}[f](x)=\sum_{n=0}^{2N}\varphi_{n,m}(x)f_{n,m}^ {1}\\ +2\sum_{n=0}^{N-1}\varphi_{2n+1,m}(x)\sum_{j=1}^{2m}\varphi_{2n+1,m}\bigg{(}1+\frac{j}{N}\bigg{)}\sum_{\ell=1}^{m}\omega_{2N+2\ell-1}^{-N-1}V_{ \ell,j}^{-1}f_{2N+2\ell-1,m}^{1}\\ -2\sum_{n=0}^{N}\varphi_{2n,m}(x)\sum_{j=1}^{2m}\varphi_{2n,m} \bigg{(}1+\frac{j}{N}\bigg{)}\sum_{\ell=1}^{m}\omega_{2N+2\ell}^{-N-1}V_{\ell,j}^{-1}f_{2N+2\ell,m}^{1}, \tag{62}\]  
where  
\[f_{n,m}^{1}:=\frac{2}{N_{m}}\sum_{k=-N}^{N}f\bigg{(}\frac{k}{N}\bigg{)}\overline {\varphi}_{n,m}\bigg{(}\frac{k}{N}\bigg{)}. \tag{63}\]  
We can prove that \(I_{N,m}^{1}[f]\) interpolates \(f\) at the grid points \(x_{k}=k/N\), \(|k|\leq N\). For even functions, the QP interpolations for the system \(\mathcal{H}_{1}\) coincide with the QP interpolations for the system \(\mathcal{H}_{class}\). Any function can be represented as a sum of even and odd functions, making enough to consider only odd ones. For odd functions \(\tilde{f}_{2n,m}=0\), \(n=0,\ldots,N+m\), and  
\[\mathcal{I}_{N,m}^{1}[f](x)=\sum_{n=0}^{N-1}F_{n,m}\sin\frac{\pi(2n+1)\sigma x }{2}, \tag{64}\]  
where  
\[F_{n,m}:=\tilde{f}_{n,m}^{1}-\sum_{\ell=1}^{m}\Theta_{n,\ell}f_{N+\ell-1,m}^ {1}=\tilde{f}_{n,m}^{1}-\sum_{\ell=m+1}^{2m}\Theta_{n,\ell}\tilde{f}_{N+\ell, m}^{1}, \tag{65}\]  
\[\tilde{f}_{n,m}^{1}=\frac{2}{N_{m}}\sum_{k=-N}^{N}f\bigg{(}\frac{k}{N}\bigg{)} s_{2n+1,k}, \tag{66}\]  
with  
\[s_{n,k}(N,m):=s_{n,k}:=\sin\pi\frac{nk}{N_{m}}, \tag{67}\]  
and  
\[\Theta_{n,\ell}(N,m):=\Theta_{n,\ell}:=2i\beta_{\ell}^{-N-1}\sum_{j=1}^{2m}s_ {2n+1,j+N}V_{\ell,j}^{-1}. \tag{68}\]  
**Theorem 4**. : _Let \(f\in C[-1,1]\) be an odd function.