## 3 Quasi-Periodic Interpolations for the Modified Trigonometric System
Hence, we can extend the system (38) by adding some new unknowns  
\[\sum_{k=-N_{m}+1}^{N_{m}-1}\frac{1}{2}\Big{[}(-1)^{n}\omega_{n}^ {k}+\omega_{-n}^{k}\Big{]}a_{k}^{*}(x)+\epsilon_{n}(x)\\ \equiv\frac{1}{2}\Big{[}(-1)^{n}e^{\frac{(n\alpha x)}{2}}+e^{- \frac{(n\alpha x)}{2}}\Big{]},\quad|n|\leq N_{m}-1, \tag{39}\]  
where  
\[\epsilon_{n}(x)\equiv 0,\quad|n|\leq 2N, \tag{40}\]  
and  
\[a_{k}^{*}(x)\equiv \begin{cases}a_{k}(x),\quad k=-N,\dots,N,\\ 0,\qquad\quad|k|>N.\end{cases}  \tag{41}\]  
Then, we multiply the both sides of (39) by \(\omega_{n}^{j}\) for \(|j|\leq N+2m\) and sum over \(n\):  
\[\sum_{k=-N_{m}+1}^{N_{m}-1}\frac{a_{k}^{*}(x)}{2}\sum_{n=-N_{m}+1 }^{N_{m}-1}\Big{[}(-1)^{n}\omega_{n}^{k+j}+\omega_{-n}^{k-j}\Big{]}+\sum_{n=- N_{m}+1}^{N_{m}-1}\omega_{n}^{j}\epsilon_{n}(x)\\ =\sum_{n=-N_{m}+1}^{N_{m}-1}\frac{\omega_{n}^{j}}{2}\Big{[}(-1)^ {n}e^{\frac{(n\alpha x)}{2}}+e^{-\frac{(n\alpha x)}{2}}\Big{]}. \tag{42}\]Based on the relations  
\[\sum_{n=-N_{m}+1}^{N_{m}-1}\left[(-1)^{n}\omega_{n}^{k+j}+\omega_{-n}^{k-j} \right]= \begin{cases}2N_{m},&k=j,\\ 2N_{m},&|k+j|=N_{m},\\ 0,&|k+j|\
eq N_{m},k\
eq j,\end{cases}  \tag{43}\]  
we obtain  
\[N_{m}a_{j}^{*}(x)+\sum_{n=2N+1}^{N_{m}-1}\omega_{n}^{j}e_{n}^{*}( x)+\sum_{n=-N_{m}+1}^{-2N-1}\omega_{n}^{j}e_{n}^{*}(x)\\ =\sum_{n=-2N}^{2N}\frac{\omega_{n}^{j}}{2}\left[(-1)^{n}e^{\frac{ i\pi mx}{2}}+e^{-\frac{i\pi mx}{2}}\right],\quad|j|\leq N+2m, \tag{44}\]  
where  
\[\epsilon_{n}^{*}(x)=\varepsilon_{n}(x)-\frac{1}{2}\Big{[}(-1)^{n}e^{\frac{ i\pi mx}{2}}+e^{-\frac{i\pi mx}{2}}\Big{]},\quad|n|=N+1,\ldots,N+2m. \tag{45}\]  
Taking into account (41), we rewrite (44) as follows:  
\[\sum_{n=2N+1}^{N_{m}-1}\omega_{n}^{j}e_{n}^{*}(x)+\sum_{n=-N_{m}+ 1}^{-2N-1}\omega_{n}^{j}e_{n}^{*}(x)\\ =\sum_{n=-2N}^{2N}\frac{\omega_{n}^{j}}{2}\Big{[}(-1)^{n}e^{\frac {i\pi mx}{2}}+e^{-\frac{i\pi mx}{2}}\Big{]},\quad|j|=N+1,\ldots,N+2m. \tag{46}\]  
From (39) and (45), we have  
\[e_{-n}^{*}(x)=(-1)^{n}e_{n}^{*}(x). \tag{47}\]  
For negative \(j\)-es, the equations of (46) can be written in the following form:  
\[\sum_{n=2N+1}^{2N+2m}(-1)^{n}\omega_{n}^{j}e_{n}^{*}(x)+\sum_{n=- 2N-2m}^{-2N-1}(-1)^{n}\omega_{n}^{j}e_{n}^{*}(x)\\ =\sum_{n=-2N}^{2N}\frac{\omega_{n}^{j}}{2}\Big{[}e^{\frac{i\pi mx }{2}}+(-1)^{n}e^{-\frac{i\pi mx}{2}}\Big{]},\quad j=N+1,\ldots,N+2m. \tag{48}\]  
After adding the equations of (48) to the corresponding equations of (46), we will obtain a system for unknowns \(\epsilon_{n}^{*}(x)\) having only even indices:  
\[\sum_{n=N+1}^{N+m}\omega_{2n}^{j}e_{2n}^{*}(x)+\sum_{n=-N-m}^{-N- 1}\omega_{2n}^{j}e_{2n}^{*}(x)\\ =\sum_{n=-N}^{N}\frac{\omega_{2n}^{j}}{2}\Big{[}e^{i\pi mx}+e^{-i \pi mx}\Big{]},\ j=N+1,\ldots,N+2m. \tag{49}\]  
Taking into account the relations  
\[\sum_{n=-N-m}^{-N-1}\omega_{2n}^{j}e_{2n}^{*}(x)=\sum_{n=N+m+1}^{N+2m}\omega_ {2n}^{j}e_{2n-2N_{m}}^{*}(x),\ j=N+1,\ldots,N+2m, \tag{50}\]  
we obtain the following linear system of equations:  
\[\sum_{\ell=1}^{2m}v_{j,\ell}b_{2N+2\ell}(x)=\sum_{n=-N}^{N}\frac{\omega_{2n}^ {j}}{2}\Big{[}e^{i\pi mxx}+e^{-i\pi mxx}\Big{]},\quad j=1,\ldots,2m, \tag{51}\]with the Vandermonde matrix (20) and unknowns  
\[b_{2n}(x):= \begin{cases}\omega_{2n}^{N+1}e_{2n}^{*}(x),&n=N+1,\ldots,N+m,\\ \\ \omega_{2n}^{N+1}e_{2n-2N_{m}}^{*}(x),&n=N+m+1,\ldots,N+2m.\end{cases}