Finding sequences in $L^2$ with specific properties Let $f \in L^2(\mathbb{R})$ and consider the set $S = \{e^{2 \pi i nx}f(x-m)\}_{n,m \in \mathbb{Z}} = \{g_{nm}(x)\}_{n,m \in \mathbb{Z}}.$ I want to find $f$ that will make this collection have certain properties. I am interested in when this sequence is complete and/or minimal. Give $f \in L^2(\mathbb{R})$ such that

*

*$S$ is complete and minimal.

*$S$ is complete and not minimal.

Just as a reminder, complete means that if $\langle h, g_{nm} \rangle = 0$, then $h = 0$. Minimal means in this context.
For the first problem I think $f = \chi_{[0,1]}$ does the trick, but what about if I want to find $f$ such that $f \neq 0$ a.s.? For the second part, I am really unsure and would appreciate any pointers.
Thanks :)!
 A: Define $e_n(x) := e^{2\pi i n x}$ and then define the operator $E_n(f) = e_nf$ (sometimes called the modulation operator). Define the translation operator $T_m(f)(x) = f(x - m) $. So your looking for a system $\{E_nT_m f \} $ that satisfies $1$ and $2$.
For $f = \chi_{[0,1]} $, the set $\{E_nT_mf\} $ is actually an orthogonal basis for $ L^2(\mathbb{R})$ (orthogonal bases are necessarily complete and minimal). Since the Fourier transform is unitary, we get that $\widehat{E_nT_mf}$ is also an orthogonal basis. But by Fourier transform symmetries, we have $\widehat{E_nT_mf} = E_mT_n\widehat{f} $. Thus $\widehat{\chi_{[0,1]}}$ is also an example. We can easily compute
$$\widehat{\chi_{[0,1]}}(\omega) = \frac{e^{2\pi i \omega} - 1}{2\pi i \omega}$$
which we see is non-zero almost everywhere and satisfies $1$.
As for $2$, that's harder and it may be that there are none. I know that if $\{E_nT_m f  \} $ forms a frame for $L^2(\mathbb{R}) $ then it is necessarily a basis (a Riesz Basis) which would make it minimal. So you need to look for cases where $ \{E_nT_m f \}$ does not form a frame.
The likeliest answer is $g(x) = e^{-\pi x^2} $. We actually do have that $\{E_nT_m g\} $ is complete and it's not a frame. I am not sure if it's minimal.
