Can we find an explicit sequence $\{f_n\}$ of unit 2-normed Schwartz functions $f_n:\mathbb R\to \mathbb C$; i.e. they lie in the unit ball of $L^2(\mathbb R)$, such that for some $\omega\in \mathbb R$ the limit \begin{align} \lim_{n\to \infty} \int_{-\infty}^{\infty}\text{d}t\; \text{e}^{-i\omega t} f_n(t)=\infty. \end{align} For example, for $\omega=0$, if a function $f:\mathbb R\to \mathbb C$ lies in $L^2(\mathbb R)$ but does not in $L^1(\mathbb R)$, then the condition follows trivially from the fact that Schwartz space is dense in $L^2(\mathbb R)$. However, this is not an explicit construction of the sequence. I am interested in finding a closed form expression for each $f_n$ that satisfies the condition mentioned.
Edit I realize now that the fact that Schwartz space is dense, does not assure that the sequence consists of function with unit norm.