I have a question about Fourier transforms:
What does the following question means?
" Does the Fourier transform of the function $f(x)= \frac{1}{\sqrt{1 + x^2}}$ belongs to $L_2(\mathbb{R})$?"
We know that $L_2(\mathbb{R})= \{ f: \mathbb{R} \rightarrow \mathbb{C} \mid \int_{- \infty}^{+ \infty} |f(t)|^2 dt < \infty \} $ and we know that $\hat{f}(t) = \int_{- \infty}^{+ \infty} e^{-itx} f(x) dx$.
Now for to show that $\hat{f}(t) \in L_2(\mathbb{R})$, what do we need to prove?
$\textbf{(1)}$ $~$ Is it enough to show that $f(x) \in L_1(\mathbb{R}) \cap L_2(\mathbb{R})$, then we have some results that shows $\hat{f}(t) \in L_2(\mathbb{R})!$
$\textbf{(2)}$ $~$ Or we have to prove $\int_{- \infty}^{+ \infty} |\hat{f}(t)|^2 dt < \infty$?
If $\textbf{(2)}$ is true? Can you please help me to show that?
Here linked-to result someone has found its Fourier transform, but I cannot understand it! I mostly prefer to find it by usual integration ways such as variable changes or substitution or even by inverse transform theorem!
Please let me know if I am wrong about using $\textbf{(1)}$ for to prove that $\hat{f}(t) \in L_2(\mathbb{R})$?
Thanks!