Proof for uniqueness of the Fourier-Transform

Question

I hav thought that the Fourier-Transform (also reverse) must be unique (under certain circumstances like integrable and continuous) in order to actually use it for signal processing. But when I am looking for proofs in books and the internet I cannot find any. So I would be happy about a proof for that :)

I just find proofs for the uniqueness of the Fourier series and its coefficients - does this already imply that the Fourier-Transform is unique?

Does the uniqueness of the DFT depend on the Nyquist-frequence (aliasing effect)?

Answer 1

Yes, you can get uniqueness for the Fourier transform from uniqueness for Fourier series, by a Poisson-summation sort of argument.

But it's very simple to do directly. Note I'm omitting the $2\pi$s:

Lemma. If $f,g\in L^1(\Bbb R)$ then $\int\hat fg=\int f\hat g$.

Proof. Fubini.

Prop. If $f\in L^1$ and $\hat f=0$ then $f=0$ a.e.

Proof. $\int f\hat g=0$ for every $g\in L^1$; in particular $\int \hat f\phi=0$ for, say, every $\phi\in C^\infty_c$.

Oops...

At the end I implicitly use the Inversion Theorem, making this somewhat circular. So instead be explicit: Say $g_{x,y}(t)=c_ye^{ixt}e^{-y|t|}$ for $y>0$ and show that $\int f\hat g_{x,y}\to f(x)$ for almost every $x$.