## 3 Quasi-Periodic Interpolations for the Modified Trigonometric System  
Initial results on the convergence of expansions using the modified Fourier system \(\mathcal{H}_{1}\) were presented in [56; 73; 74; 75]. We highlight two theorems for further comparison. **Theorem 1** ([74; 75]). : _Assume \(f\in C^{2}[-1,1]\) and \(f^{\prime\prime}\in BV[-1,1]\). If \(|x|<1\), then_  
\[f(x)-\mathcal{I}^{1}_{N}[f](x)=O(N^{-2}),\,N\to\infty. \tag{26}\]  
_Otherwise,_  
\[f(\pm 1)-\mathcal{I}^{1}_{N}[f](\pm 1)=O(N^{-1}),\,N\to\infty. \tag{27}\]  
This indicates improved convergence compared to classical Fourier expansions ([82]), with an enhancement of order \(O(N)\). This improvement is associated with faster decay of the coefficients \(\hat{f}_{n}\):  
\[\hat{f}_{n}=O(n^{-2}),\,n\to\infty, \tag{28}\]  
which results from the non-periodicity of the basis functions \(\sin\pi(n-\frac{1}{2})x\) on \([-1,1]\). Now, we consider interpolations that are exact for the eigenfunctions of the modified system \(\mathcal{H}_{1}\). Upcoming theorems prove the improved convergence of interpolations compared to the expansions \(F^{1}_{N}[f]\). The basis functions can be expressed in the complex form, which is convenient for further derivations  
\[\varphi_{n}(x):= \begin{cases}\frac{1}{\sqrt{2}},\qquad n=0,\\ \\ \frac{1}{2}\Big{[}(-1)^{n}e^{\frac{i\pi nx}{2}}+e^{-\frac{i\pi nx}{2}}\Big{]}, \qquad n\in\mathbb{N}.\end{cases}  \tag{29}\]  
We denote the corresponding interpolations \(\mathcal{I}^{1}_{N}[f](x)\), derived in [77]:  
\[\mathcal{I}^{1}_{N}[f](x):=\sum_{n=0}^{2N}\hat{f}_{n}^{1}\varphi_{n}(x), \tag{30}\]  
where  
\[\hat{f}_{n}^{1}=\frac{2}{2N+1}\sum_{k=-N}^{N}f(x_{k})\overline{\varphi}_{n}(x _{k}),\,\,x_{k}=\frac{2k}{2N+1}. \tag{31}\]  
Define  
\[A_{s}(f):=A_{s}:=f^{(s)}(1)+(-1)^{s-1}f^{(s)}(-1). \tag{32}\]  
**Theorem 2** ([77]). : _Let \(f\) be an odd function on \([-1,1]\). Assume that \(f\in C^{4}[-1,1]\) and \(f^{(4)}\in BV[-1,1]\). Then, the following estimate holds for \(|x|<1\):_  
\[f(x)-\mathcal{I}^{1}_{N}[f](x)=A_{1}(f)\frac{(-1)^{N}\pi}{32N^{3}}\frac{\sin \pi(N+\frac{1}{2})x}{\cos^{2}\frac{\pi x}{2}}+O(N^{-4}),\,N\to\infty. \tag{33}\]  
**Theorem 3** ([77]). : _Let \(f\) be an odd function on \([-1,1]\). Assume that \(f\in C^{2}[-1,1]\) and \(f^{(2)}\in BV[-1,1]\). Then, the following estimate holds:_  
\[f(\pm 1)-\mathcal{I}_{N}^{1}[f](\pm 1)=\pm A_{1}(f)\frac{(-1)^{N+1}}{2\pi N}+o(N^ {-1}),\,N\to\infty. \tag{34}\]  
Comparing Theorem 1 with Theorems 2 and 3 suggests that interpolations provide better convergence within the interpolation interval, away from the endpoints. We now focus on QP interpolations for the system \(\mathcal{H}_{1}\) and demonstrate their improved convergence compared to both \(\mathcal{I}_{N}^{1}[f]\) and \(\mathcal{F}_{N}^{1}[f]\). Following the QP interpolation concept, we modify the functions \(\varphi_{n}(x)\) to \(\varphi_{n,m}(x)\):  
\[\varphi_{n,m}(x):= \begin{cases}\frac{1}{\sqrt{2}},\qquad n=0,\\ \frac{1}{2}\Big{[}(-1)^{n}e^{\frac{(n\alpha x)}{2}}+e^{-\frac{(n\alpha x)}{2} }\Big{]},\qquad n=1,\dots,2N.\end{cases}  \tag{35}\]  
We seek the QP interpolation for \(\mathcal{H}_{1}\) in the form  
\[\mathcal{I}_{N,m}^{1}[f](x):=\sum_{k=-N}^{N}f\bigg{(}\frac{k}{N}\bigg{)}a_{k} (x), \tag{36}\]  
where the coefficient functions \(a_{k}(x)\) are determined by requiring that \(\mathcal{I}_{N,m}^{1}[f](x)\) is exact for the basis functions \(\varphi_{n,m}\):  
\[\mathcal{I}_{N,m}^{1}[\varphi_{n,m}](x)\equiv\varphi_{n,m}(x),\,\,n=0,1,\dots, 2N. \tag{37}\]  
We have  
\[\sum_{k=-N}^{N}\frac{1}{2}\Big{[}(-1)^{n}\omega_{n}^{k}+\omega_{-n}^{k}\Big{]} a_{k}(x)\equiv\frac{1}{2}\Big{[}(-1)^{n}e^{\frac{(n\alpha x)}{2}}+e^{-\frac{(n \alpha x)}{2}}\Big{]},\quad n=0,1,\dots,2N. \tag{38}\]  
Taking into account that \(\varphi_{-n,m}(x)=(-1)^{n}\varphi_{n,m}(x)\), we see that (38) is exact also for \(\varphi_{-n,m}(x)\), \(n=1,\dots,2N\).