Suppose $f\in L^1(\mathbb{R})$. Define the Fourier transform of $f$ as:
$$\hat{f}:\mathbb{R}\rightarrow\mathbb{C}, \xi\mapsto\int_{\mathbb{R}}f(t)e^{-2\pi i\xi t}\operatorname{d}t.$$
Suppose that $\hat f\in L^1(\mathbb{R})$.
Under what further hypothesis on $f$ and $\hat{f}$can we make rigorous the following argument:
$$\forall x\in\mathbb{R}, \int_\mathbb{R} \hat f(\xi)e^{2\pi i\xi x}\operatorname{d}\xi \overset{(1)}{=} \lim_{\Delta L\rightarrow0^+} \sum_{n=-\infty}^{+\infty} \left(\hat{f}(n\Delta L)e^{2\pi \Delta L in x}\Delta L\right) \\ = \lim_{\Delta L\rightarrow0^+} \sum_{n=-\infty}^{+\infty} \left(\left(\int_{\mathbb{R}}f(t)e^{-2\pi\Delta L i n t}\operatorname{d}t\right)e^{2\pi in \Delta Lx}\Delta L\right) \\ = \lim_{\Delta L\rightarrow0^+} \sum_{n=-\infty}^{+\infty} \left(\int_{\mathbb{R}}f(t)e^{2\pi\Delta L i n (x-t)}\operatorname{d}t \Delta L\right) \\ = \lim_{\Delta L\rightarrow0^+} \sum_{n=-\infty}^{+\infty} \left(\sum_{k=-\infty}^{+\infty}\int_{\frac{1}{\Delta L}[k,k+1]}f(t)e^{2\pi\Delta L i n (x-t)}\operatorname{d}t \Delta L\right) \\ = \lim_{\Delta L\rightarrow0^+} \sum_{n=-\infty}^{+\infty} \left(\sum_{k=-\infty}^{+\infty}\int_{\frac{1}{\Delta L}[k,k+1]}f(t)e^{2\pi\Delta L i n \left((x+\frac{k}{\Delta L})-t\right)}\operatorname{d}t \Delta L\right) \\ \overset{(2)}{=} \lim_{\Delta L\rightarrow0^+} \sum_{k=-\infty}^{+\infty} \left(\sum_{n=-\infty}^{+\infty}\int_{\frac{1}{\Delta L}[k,k+1]}f(t)e^{2\pi\Delta L i n \left((x+\frac{k}{\Delta L})-t\right)}\operatorname{d}t \Delta L\right) \\ = \lim_{\Delta L\rightarrow0^+} \sum_{k=-\infty}^{+\infty} \left(\sum_{n=-\infty}^{+\infty} \left(\frac{1}{\frac{1}{\Delta L}}\int_{\frac{1}{\Delta L}[k,k+1]}f(t)e^{-2\pi\Delta L i n t}\operatorname{d}t\right)e^{2\pi \Delta L in(x+\frac{k}{\Delta L})}\right) \\ \overset{(\star)}{=} \lim_{\Delta L\rightarrow0^+} \sum_{k=-\infty}^{+\infty} f\left(x+\frac{k}{\Delta L}\right)\overset{(3)}{=}f(x) ?$$
The idea is to prove Fourier inversion formula in $\mathbb{R}$ using the Fourier inversion formula on the torus, that we used in $(\star)$.
In particular, while it seems that we can take care of $(1)$ and $(3)$ imposing some decreasing condition at $\infty$ on $f$ and $\hat{f}$, I'm having hard times in finding hypothesis on $f$ and $\hat{f}$ for which $(2)$ holds, neither I have an idea on what technique I could use to prove it.
Best Answer
The answer provided by David C. Ulrich in this post can be adapted to answer positively this question under the hypothesis in there.