Orthonormal system for $L^2(\mathbb R)$

Solution 1:

Define the function $f(x)$ by

$$f(x) = \sum_{m\in\mathbb{Z}}\vert \phi(x-m)\vert^2-1\ .$$

Then it is easy to see $f$ is $1$-periodic and integrable on $[0,1]$. Then the Fourier coefficients $\hat f(k)$ are defined

$$\hat f(k) = \int_0^1 e^{2\pi i k x}f(x) dx\ ,$$

Also define the collection $\phi_n(x)=e^{2\pi i n x}\phi(x)$ for each $n\in\mathbb{Z}$. We can calculate

$$\begin{align}\hat f(k) &= \int_0^1e^{2\pi i kx }\left[\sum_{m\in\mathbb{Z}}\vert \phi(x-m)\vert^2-1\right]dx \\ &=\sum_{m\in\mathbb{Z}}\int_0^1 e^{2\pi i kx }\vert \phi(x-m)\vert^2dx -\delta_{k,0}\\ &=\sum_{m\in\mathbb{Z}}\int_{-m}^{-m+1} e^{2\pi i k(x+m) }\vert \phi(x)\vert^2dx -\delta_{k,0}\\ &=\int_{-\infty}^{\infty} e^{2\pi i kx }\vert \phi(x)\vert^2dx - \delta_{k,0} \ .\end{align}$$

Which gives us

$$\hat f(m-n) = \langle \phi_m,\phi_n\rangle - \delta_{m,n}$$

So we have that $\hat f(k)= 0$ for every $k\in\mathbb{Z}$ if and only if $\{\phi_n(x)\}$ is an orthonormal system. However, we also know that $\hat f(k) = 0$ for every $k$ if and only if $f(x)=0$ for a.e. $x\in [0,1]$.