I am using multiple textbooks on Fourier Series, and the only one that contains a proof is Mark A. Pinsky's Introduction to Fourier Analysis and Wavelets. However, since it is so concise, I am asking for an explanation of this proof.
Here is the main part of the Proof:
We first show that the partial sums converge in the norm of $L^2$. We have $$\left\lVert S_Nf-S_Mf \right\lVert^2 = \frac{1}{2\pi} \int_T | S_Nf - S_Mf|^2 =^{(*_1)} \sum _{M+1}^N |\hat{f}(n)|^2 \rightarrow 0$$ as $M$, $N \rightarrow \infty$. by the Bessel inequality. Since $L^2$ is complete, we show that $F = \lim_NS_Nf = f$ by writing $$2\pi \widehat{F}(n) = \int_T[F(\theta)-S_Nf(\theta)]e^{-in\theta} + \int_T S_Nf(\theta)e^{-in\theta} d\theta$$ If $N > |n|$, the last integral $= 2\pi \hat{f}(n)^{(*_2)}$. Then apply Cauchy-Schwartz...
Could you explain in detail how $(*_1)$ and $(*_2)$ works?
For $(*_1)$I tried to write the difference of partial sums out, but don't know how to deal with the norm or the square or why Bessel inequality implies this convergence.
For $(*_2)$ I tried again to write out the partial sum but couldn't see the equality here or understand what $N > |n|$ implies.
Thank you!
Both of them are consequences of orthogonality of $\{e^{inx}\}$, i.e. $$\frac{1}{2\pi}\int_Te^{inx}e^{-imx}dx=\delta_{n,m}=\begin{cases}1, &n=m; \\ 0, &n\ne m.\end{cases}$$ For example, in ($\ast_1$), $$\|S_Nf-S_Mf\|^2=\left\|\sum_{n=M+1}^N\hat{f}(n)e^{inx}\right\|^2=\sum_{n=M+1}^N|\hat{f}(n)|^2$$ by Pythagorean. Can you see why ($\ast_2$) is true now?