Let $f \in C_c^0(\mathbb{R})$ be a continuous function with compact support. Is it true that
$f_n(x)=f(-x+n^3)\rightharpoonup0$ in $L^1(\mathbb{R})$?
$f_n(x)=f(x+\frac{1}{e^n})\rightharpoonup0$ in $L^1(\mathbb{R})$?
I know that in order to determine if a sequence $f_n$ in $L^p$ weakly converges to a function $f$ in $L^p$ I have to show if $$\int f_n(x)g(x)dx \longrightarrow \int f(x)g(x) dx \quad \forall g\in L^{p'}$$
Is it true that, for case 1., if I take $f_n(x)= \chi_{[-n,n]}*\rho_{\epsilon}$ I obtain a continuous function with compact support and $\int f_n(x)g(x)dx \not \longrightarrow 0 \quad \forall g\in L^{p'}$ because the translation by $n^3$ doesn't affect the behaviour at infinity?
I don't understand what are the differences between the two cases 1. and 2. and how to deal with translations.
Have you any hint?
Hints$\newcommand{\R}{\mathbb{R}}$
For question 1: since $L^{1'}(\R) = L^{\infty}(\R)$, we can take $g \equiv 1$ in $L^{\infty}(\R)$. Then remark that: $$ \int f_n \cdot g = \int f_n = \int f $$
For question 2: what function does $(f_n)_n$ converge pointwise to? What do the norms of the $f_n$ converge to?