PROBLEM (1)$f \in L_1 (\mathbb R)$ so that $xf(x), x \hat{f}(x) \in L_1 (\mathbb R)$, show that $\sum|\hat{f}(n)| \le \infty$.
(2)Also, show that $\sum_{n\in \mathbb Z} f(x-2\pi n)$ converges uniformly on $[-\pi, \pi]$
(3)Finally, prove the Poisson summation formula: for every $x \in > \mathbb R$ $$\sum_{-\infty}^{\infty} f(x+2 \pi n) = 1/2 \pi > \sum_{-\infty}^{\infty}\hat f (n)e^{inx} $$
I even obstruct by (1) and cannot continue. Using the condition, Fourier inversion and dominated convergence , $xf(x), x \hat{f}(x) \in L_1 (\mathbb R)$, I can show successively $\hat{f} \in L_1 (\mathbb R)$ , $f \in C_0 (\mathbb R)$ and $f \in C_0^1 (\mathbb R)$ Symmetrically, I also can show $\hat f \in C_0^1 (\mathbb R)$.
Provided $f' \in L_1 (\mathbb R)$, we will have: $$ \hat f (n) =\int_{\mathbb R}\,f(x)e^{-i n x} dx = \frac{f(x)e^{-i n x}}{-inx} \Big |^{\infty}_{-\infty}+(in)^{-1}\int_{\mathbb R}\,f'(x)e^{-i n x} dx =(in)^{-1} \widehat{f'}(n) $$ I try to imitate the proof of the corresponding question regarding $L_1 (\mathbb T)$, however, it doesn't work since we don't have $f' \in L_1 (\mathbb R)$. Please help!