Sequence of functions
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Sequence of functions
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 \(\displaystyle{\sum_{k=1}^{\infty}\dfrac{1}{2\,\pi}\,\int_{-\pi}^{\pi}\left|f_{k^2}(x)\right|\,\mathrm{d}x=\sum_{k=1}^{\infty}\dfrac{1}{k^2}}\).
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 \(\displaystyle{\sum_{k=1}^{\infty}\dfrac{1}{2\,\pi}\,\int_{-\pi}^{\pi}\left|f_{k^2}(x)\right|\,\mathrm{d}x=\sum_{k=1}^{\infty}\dfrac{1}{k^2}}\).
Re: Sequence of functions
May we have a hint Vaggelis?
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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- Community Team
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Re: Sequence of functions
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
\(\displaystyle{||x_1+...+x_n||^2=||x_1||^2+...+||x_n||^2}\).
such that \(\displaystyle{\langle{x_i,x_j\rangle}=0\,,\forall\,i\,,j\in\left\{1,...,n\right\}\,,i\neq j}\), then
\(\displaystyle{||x_1+...+x_n||^2=||x_1||^2+...+||x_n||^2}\).
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