Sequence of functions

[Papapetros Vaggelis]

Let \(\displaystyle{\left(b_{n}\right)_{n\in\mathbb{N}}}\) be a strictly increasing sequence of natural

numbers. For every \(\displaystyle{m\in\mathbb{N}}\), we deifne

\(\displaystyle{f_{m}(x)=\dfrac{1}{m}\,\sum_{n=1}^{m}e^{i\,b_n\,x}\,\,,x\in\left[-\pi,\pi\right]}\).

Prove that \(\displaystyle{\sum_{k=1}^{\infty}\dfrac{1}{2\,\pi}\,\int_{-\pi}^{\pi}\left|f_{k^2}(x)\right|\,\mathrm{d}x=\sum_{k=1}^{\infty}\dfrac{1}{k^2}}\).

[Riemann]

May we have a hint Vaggelis?

[Papapetros Vaggelis]

If \(\displaystyle{\left(H,\langle{,\rangle}\right)}\) is a Hilbert space and \(\displaystyle{x_1,...,x_n\in H}\)

such that \(\displaystyle{\langle{x_i,x_j\rangle}=0\,,\forall\,i\,,j\in\left\{1,...,n\right\}\,,i\neq j}\), then

\(\displaystyle{||x_1+...+x_n||^2=||x_1||^2+...+||x_n||^2}\).