Doubt in the definition of the Fourier transform in $L^{2}(\mathbb R^n)$

Question

I am trying to understand the definition of the Fourier transform in $L^{2}(\mathbb R^n)$ .

I am understand of this manner :

Let $f \in L^{2}(\mathbb R^n)$ and $n$ a natural number. Define $f_n = f {\chi}_{B(0,n)}$. The Fourier transform of $f$ is the function $\hat{f}$ given by

$$ \hat{f} = \lim_n \hat{f_n} \ \ \text{(the limit is in $L^2{(\mathbb R^n)}$)}$$

and the Fourier transfom of $f$ in a point $\xi$ is given by

$$ \hat{f}(\xi) = \lim_{n} \int_{B(0,n)} f_n(x)e^{-2 \pi i \langle x,\xi\rangle} \ dx$$

I am right?

Thanks in advance

Answer 1

This is correct.

The reason you had to do all these stuff is the fact that $L^2$-function on $\mathbb R^n$ are not necessarily integrable (i.e. $L^1$), but they can be approximated in the $L^2$-norm by functions which are in $L^1\cap L^2$, and this is precisely what you are doing.