For $f$ and $f'$ in $L^2(0,1)$, define $e_k(x)=e^{2\pi ikx}$, $k \in \mathbb{Z}$. And define the Fourier series: $f=\sum _{k \in \mathbb{Z}}c_ke_k$, where $c_k=\left \langle f,e_k \right \rangle=\int_{0}^{1}f(x)e^{-2\pi ikx}dx$. Also suppose $f(x)=\sum _{k \in \mathbb{Z}}\left \langle f,e_k \right \rangle e_k(x)$, $\forall x\in [0,1]$.
My question is how to show this relationship: $|f(x)-\sum _{k \leq |N|}\left \langle f,e_k \right \rangle e_k(x)|\leq \frac{1}{\sqrt{2}\pi}\frac{1}{\sqrt{N}}(\int_{0}^{1}|f'(t)|^2dt)^{1/2}$. (For all $n\in \mathbb{N}$)
I think we will use the fact that $|e^{in}|=1$ and C-S inequality but still get stuck.
This formula is important because we can estimate the number of terms we have to keep in the partial sum of the Fourier series in order to obtain a given pointwise approximation of $f(x)$. But how to prove it?
Thanks in advance!
Use that, for 1-periodic $C^1$ functions, $⟨f',e_k⟩=-⟨f,e_k'⟩=2\pi i\cdot k\cdot⟨f,e_k⟩$ and then $$ \sum_{k=N+1}^\infty\frac1{k^2}\le\sum_{k=N+1}^\infty\frac1{k^2-1}=\frac1{2N}+\frac1{2N+2}<\frac1{N} $$