Consider $f,\mathscr F f \in L^1(\mathbf R^n)$. Dominated convergence theorem implies that $$\mathscr F[f(x)]\cdot \mathbf 1_{|x|\le R}\overset{L_1}{\longrightarrow} \mathscr F[f(x)]$$ But what can be said about $$g_R=\mathscr F[f(x)\mathbf 1_{|x|\le R}]$$ Definitely $g_R\in L^1$ but do these functions converge to $\mathscr F[f]$ in $L_1$?
If $f\in L^2$ it's well-known that $\mathscr F[f(x)\mathbf 1_{|x|\le R}]\overset{L_2}{\longrightarrow}\mathscr F[f].$
I believe that these functions do not have to converge to Fourier transform, so I should construct a counterexample, but I struggle with it, so any hint is appreciated! Thanks!
P.S. I think the example shoud also be 1-dimensional, but trivial attempt to check exponents like $e^{-x^2} $ didnt work, as soon as integral over finite interval has no elementary formula.