Consider the space $X = L^2(\mathbb{R})$. For $f \in X$ and $\theta > 0$ define:
$$ \| f \|_\theta := \int^\infty_{-\infty} e^{-2 \theta |x|}| \hat f(x) |^2 \, dx, $$
and
$$ \| f \|_{X_\theta} := \sup_{a \in \mathbb{R}} \|f \chi_{(-\infty, a)}\|_\theta . $$
Question: Take $\alpha > 0$. Then, for $\theta > \alpha$, clearly:
$$ \| f \|_{X_\theta} \leq \| f \|_{X_\alpha}. $$
But is it true that
$$ \limsup_{\theta \rightarrow \alpha+} \|f\|_{X_\theta} = \|f\|_{X_\alpha}? $$