I have a question concerning the upper bound of the sum of the Fourier series and how to prove the following.
I have the sum as follows:
$$\int^\pi_{-\pi} |\frac{1}{n} \sum_{k = m-n}^{m-1} S_k|^2 \leq \int_{-\pi}^{\pi} |f|^2$$
where $ f \in L^2[-\pi, \pi]$ with $m,n \in \mathbb{N} , m>n$, and
$$S_p = \sum_{k = -p}^{p} \hat f_k e^{ikx}$$
where $\hat{f}_k$ is the $k$'th Fourier series coefficient defined as
$$\hat f_k = \frac{1}{2\pi}\int^{\pi}_{-\pi} f(y) e^{-iky} dy$$
I've also tried using the fact that $e^{ikx}$ forms a basis in $L^2$ but I haven't achieved much
edit: changed $c_k$ to $\hat f$
Best Answer
First note that $$ \int_{-\pi}^\pi |S_k|^2\,dx=2\pi\sum_{j=-k}^{k}|\hat{f}_k|^2\le 2\pi\sum_{j=-\infty}^{\infty}|\hat{f}_k|^2\le \int_{-\pi}^\pi |f|^2\,dx, $$ Then, using the fact that $$ |a_1+\cdots+a_n|^2\le n(|a_1|^2+\cdots+|a_n|^2) $$ we obtain $$ \frac{1}{n^2}\int_{-\pi}^\pi \left|\sum_{k=m-n}^{m-1}S_k\,\right|^2\,dx \le \frac{1}{n}\sum_{k=m-n}^{m-1}\int_{-\pi}^\pi |S_k|^2\,dx\le \int_{-\pi}^\pi |f|^2\,dx. $$
Proof of the Fact: $$ |a_1+\cdots+a_n|^2\le (|a_1|+\cdots+|a_n|)^2=\sum_{i,j=1}^n|a_i||a_j|\\ \le \frac{1}{2}\sum_{i,j=1}^n(|a_i|^2+|a_j^2|)= n(|a_1|^2+\cdots+|a_n|^2) $$ $$ $$