Orthonormal Basis of $L^{2}(0,1)$?

Question

Are the functions $e_n := e^{i\cdot(2n+1)\cdot\pi\cdot x}$, $n \in \mathbb{Z}$ an orthonormal basis of $L^{2}(0,1)$? I suppose it is true, but I haven't been able to prove it myself yet.

Answer 1

I have found a proof. It makes use of the following facts:

• $\{e^{i\cdot 2\pi nx}: n \in \mathbb{Z}\}$ is an orthonormal basis of $L^{2}(0,1)$.
• Let $\{e_k : k \in I \}$ be an orthonormal set in a Hilbert Space H and let M denote the closure of its span. Then, for $ x \in H$, the following two statements are equivalent:
  1. $ x \in M$
  2. $\sum_{k\in I}|(x,e_k )|^2 = \|x\|^2$
• $\sum_{n\in \mathbb{N}}\frac{1}{n^2} = \frac{\pi^{2}}{6}$

Let M denote the closure of the span of the set $\{e^{i\cdot (2n+1)\pi x}: n \in \mathbb{Z}\}$. Let $\tilde e_{l} := e^{i\cdot 2\pi lx}$ for $l \in \mathbb{Z}$. We show that M is dense in H using the statements above: It suffices to check $\sum_{k \in \mathbb{Z}} |(\tilde e_l,e_k)|^2 = 1 $ for all $l \in \mathbb{Z}$. For all $l \in \mathbb{Z}$, a straightforward calculation yields

$$\sum_{k \in \mathbb{Z}} |(\tilde e_l,e_k)|^2 = \frac{4}{\pi^2} \sum_{k \in \mathbb{Z}} \frac{1}{(2k+1)^2} $$

We can now find the value of this series by observing

$$\sum_{n\in \mathbb{N}} \frac{1}{n^2} = \sum_{n\in \mathbb{N}}^{} \frac{1}{(2n)^2} + \sum_{n\in \mathbb{N}} \frac{1}{(2n - 1)^2}$$

Therefore, $$ \sum_{n\in \mathbb{N}} \frac{1}{(2n - 1)^2} = \frac{3}{4}\cdot \sum_{n\in \mathbb{N}} \frac{1}{n^2} = \frac{\pi^2}{8}$$

Using the identity $\sum_{k \in \mathbb{Z}} \frac{1}{(2k+1)^2} = 2 \cdot \sum_{n\in \mathbb{N}} \frac{1}{(2n - 1)^2}$ we obtain $\sum_{k \in \mathbb{Z}} |(\tilde e_l,e_k)|^2 = 1 $ for all $l \in \mathbb{Z}$.