I try to identify conditions for the Fourier-transformation $\mathcal{F}(f)$ of some function $f \in L^1(\mathbb{R}^n)$ to be real-analytic. Namely I want to show that one of the following two conditions is sufficient:
- $\exists K \text{ compact}: \text{supp}(f) \subset K$
- $\exists C\in (0,\infty): |f|\leq\exp(-C|x|)$
A few notes:
Certainly, what we are looking for is a good bound on the derivatives, in particular $|D^\alpha\mathcal{F}(f)|\leq C_n/r^{|\alpha|}$ where $\alpha$ is a multi-index would be some inequality of the form I am interested in.
I tried to find some resources and found the Paley-Wiener theorems, but I could not find any resource that shows what I am looking for. Rudin shows in "Functional Analysis" (Theorem 7.22+) that if $f$ is smooth and has compact support $\mathcal{F}(f)$ is holomorphic and the restriction of it to the reals is real-analytic then. However, I want to find a proof that doesn't rely on complex-analysis and I would need to remove the restriction that $f$ is smooth.
I currently work with the following definition of the Fourier-transformation:
$\mathcal{F}(f)(t) = \int_{\mathbb{R}^n} f(x)\, e^{-2\pi \mathrm{i} t \cdot x} \,\mathrm{d} x.$