Assume you want to compute the Fourier transform of a function $f_\epsilon(x)$ given by
\begin{align} \mathcal{F}(f_\epsilon)(k) = \int f_\epsilon(x) e^{-ikx}\, dx \end{align}
Further assume, that $f_\epsilon \rightarrow f$ as $\epsilon \rightarrow 0$.
My question is:
Does \begin{align} \lim_{\epsilon \rightarrow 0 } \, \, \mathcal{F}(f_\epsilon) = \mathcal{F}(\lim_{\epsilon \rightarrow 0 }\, \, f_\epsilon) \end{align} hold true? What are the necessary assumptions on $f$ and $f_\epsilon$. It would be nice to have some references.
Thank you very much.
EDIT:
I apologize very much for not beeing specific enough. Indeed, my function behaves nicely and convergeces uniformely.
There are several ways that make this thing hold true. The first one, also the useless one, is that $f_{\varepsilon}\rightarrow f$ uniformly on $\mathbb{R}^n$. An other thing should verify the hypothesis of dominated convergence theorem. (https://en.wikipedia.org/wiki/Dominated_convergence_theorem) What I want to say is that your problem is not very much different from the problem of taking the limit under the sign of the integral. And, moreover, one has to be more precise in the statement of the last equality in your question. For example, if your function is in $L^2(\mathbb R^n)$ then your equality is just an almost everywhere equality, while if $f\in L^1(\mathbb{R}^n)$ is a pointwise equality, because its Fourier tranform must be continuous...