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Background

Consider the Hilbert space $L^2([-\pi,\pi]).$ It is well-known that the set of functions $$\left \lbrace \frac{e^{inx}}{\sqrt{2 \pi}} \right \rbrace_{n \in \mathbb {Z}}$$ is an orthonormal basis for the entire space with respect to the inner product $$\langle f(x),g(x) \rangle =\int_{-\pi}^{\pi} f(x)\overline{g(x)} \ dx.$$

Hence any $f(x) \in L^2([-\pi,\pi])$ can be written as $$f(x)=\sum_{k \in \mathbb{Z}} \alpha_k \frac{e^{ikx}}{\sqrt{2 \pi}},$$ where $\alpha_k=\left \langle f(x), \frac{e^{ikx}}{\sqrt{2 \pi}} \right \rangle.$ Further, Parseval's theorem says here that $$\int_{-\pi}^{\pi} |f(x)|^2 \ dx= \sum_{k \in \mathbb{Z}} |\alpha_k|^2.$$

An application of Parseval's Theorem to the function $f(x)=x$ is one of the many standard proofs of the Basel Problem, which one may recall is to show that $$\sum_{n \in \mathbb{N}} \frac{1}{n^2}=\frac{\pi^2}{6}.$$

Questions

Consider the orthonormal system $$\left \lbrace \frac{e^{i(3n+1)x}}{\sqrt{2 \pi}} \right \rbrace_{n \in \mathbb {Z}}.$$

Does the closed linear span of the above system (the subspace for which the system would become a basis) have any interesting or special characterization ?

Could some variant of Parseval's identity be applied to the system to prove that $$\sum_{n \in \mathbb{Z}} \frac{1}{(3n+1)^2} = \frac {4 \pi^2}{27} ?$$

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    $\begingroup$ Yes, trivially that orthonormal system is an orthonormal basis of its linear span. This is always the case, with any set of orthonormal vectors. Concerning the characterization, I have no idea. $\endgroup$ – Giuseppe Negro Apr 2 at 20:24
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    $\begingroup$ I remember doing this way back when. I think when you do, everything works out and you get something like $$ \sum_{n\in\Bbb Z}\frac1{(k n + m)^2} = \frac{\pi^2}{k^2} \csc^2\left( \frac{m\pi}{k}\right)$$. This works out nicely for $n = 1, 2, 3, 4, 5, 6$ (for example, $$ \sum_{n\in\Bbb Z}\frac1{(5 n + 2)^2} = \frac{\pi^2}{25} \frac8{5+\sqrt{5}}$$ but not so cleanly when $n=7$ or any number of sides of a non-constructible regular polygon. $\endgroup$ – Mark Fischler Apr 2 at 20:43
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If $$f(x) = \sum_n c_n e^{inx} \qquad(\text{convergence in } L^2_{loc})$$ then $$\frac{f(x)+e^{-2i\pi /3} f(x+2\pi/3)+e^{-4i\pi /3} f(x+4\pi/3)}{3} = \sum_{n\equiv 1 \bmod 3} c_n e^{inx} \qquad(\text{convergence in } L^2_{loc})$$

The LHS is then the orthogonal projection of (the periodization of) $L^2([-\pi,\pi])$ on the closure of the subspace spanned by $\left \lbrace \frac{e^{i(3n+1)x}}{\sqrt{2 \pi}} \right \rbrace_{n \in \mathbb {Z}}.$

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