Fourier transform dependent upon a parameter and $L^2$ convergence

Question

Suppose I know the Fourier transform of a function depending upon a parameter, call it $f_\epsilon(x)$, and that I want to know the Fourier transform of a function $f(x)$. Furthermore, suppose I know that $\|f-f_\epsilon\|\rightarrow0$ for $\epsilon \rightarrow0$ in the sense of the $L^2$ norm (or $L^1$). Can I recover the Fourier transform of $f(x)$ by taking the limit of the Fourier transform $\hat f(\omega,\epsilon)$ for $\epsilon \rightarrow0$?

Answer 1

If you consider $ ||f-f_\epsilon ||_1\rightarrow_{\epsilon \to 0} 0 $ then we have

$$ \Bigg| \int_{-\infty}^{\infty} f_{\epsilon} (x) e^{-ixw}dx - \int_{-\infty}^{\infty} f (x) e^{-ixw}dx \Bigg| \leq \int_{-\infty}^{\infty}|f_{\epsilon}(x)-f(x)|dx $$

$$ = ||f-f_\epsilon ||_1 \rightarrow_{\epsilon \to 0}0 .$$