In a task I should first show that $\mathcal{F}(e^{-\frac{x^2}{2 k}})=\sqrt{k} e^{-\frac{k \xi ^2}{2}}$ if $\text{Re}(k)>0$.
Then they say that one may conclude that there is a sequence $(f_k)_{k \in \mathbb{N}}$ such that $\text{ess sup}(|f_j|)=1$, $f_j \in C^\infty$ and $f_j \in L_1$ such that the $L_1$ norm of $\mathcal{F}(f_k)$ diverges to $\infty$, namely $\lim_{k \rightarrow \infty}||\mathcal{F}f_k||_1=\infty$.
My question is how to find such $f_k$. (We consider complex valued functions in the whole task). I could show the identity but I cannot understand at all how we could construct such $f_k$'s as
$||\mathcal{F}(e^{-\frac{x^2}{2 k}})||_1=\int_{-\infty }^{\infty } \sqrt{k} e^{-\frac{1}{2} \left(k \xi ^2\right)} d\xi=2\pi$
Therefore you cannot choose the $k$'s wisely to get such a sequence. Can one consider a convolution with a different function to get the result? The problem is for all examples I looked at the property $\text{ess sup}(|f_j|)=1$ was not fulfilled, I hope that I am not overlooking something trivial.