The following higher order estimate for periodic Sobolev function $ f \in H_\text{per}^r(0,2\pi) $ is a really practical result in numerical analysis.
Let $ P_n : L^2 ( 0, 2\pi ) \longrightarrow \mathcal{T}_n \subset L^2(0, 2\pi ) $ be an orthogonal projection operator, where \begin{equation*} \mathcal{T}_n := \left\{ \xi_0 \cdot \frac{1}{\sqrt{2\pi}} + \sum^n_{k=1}\xi_k \frac{\cos kt}{\sqrt{\pi}} + \sum^n_{k=1} \eta_k \frac{\sin kt}{\sqrt{\pi}} : \ \xi_0, \xi_k , \eta_k \in \mathbb{R} \right\}. \end{equation*} Then $ P_n $ is given as follows： \begin{equation} (P_n x)(t) = \xi_0 \frac{1}{\sqrt{2\pi}} + \sum^n_{ k = 1 }\xi_k \frac{\cos k t}{\sqrt{\pi}} + \sum^n_{ k=1 } \eta_k \frac{\sin k t}{\sqrt{\pi}} , \end{equation} where \begin{equation*} \xi_0 = \frac{1}{\sqrt{2\pi}} \int^{2\pi}_0 x(t) \, dt ,\ \xi_k = \int^{2\pi}_0 x(t) \frac{ \cos k t}{\sqrt{\pi}} \, dt, \end{equation*} \begin{equation*} \eta_k = \int^{2\pi}_0 x(t) \frac{ \sin k t}{\sqrt{\pi}} \, dt, \ \ k \in \overline{1,2,\ldots,n} \end{equation*} are the Fourier coefficients of $ x $. Furthermore, the following estimate holds: \begin{equation*} \Vert x- P_n x \Vert_{L^2} \leq \frac{1}{ n^r } \Vert x \Vert_{H^r} \text{ for all } x \in \ H_\text{per}^r(0,2\pi), \end{equation*} where $ r \geq 0 $.
Now I was wondering if there exist a similar estimate for a non-periodic Sobolev function $ f \in H^r(0,2\pi) $?
Here notice that the precipitation of a term $ \frac{1}{n^r} $ is the key point I mostly need.
Thanks in advance!
