Analytic signal of a real band-pass signal is stable

Question

Let $f \in L^1(\mathbb{R})$ such that $f$ is real valued and such that $\text{supp} \hat{f}$ is compact, where $$ \hat{f}(x) = \int_{\mathbb{R}} f(t)e^{-2\pi i tx} dt $$ is the Fourier transform of $f$. The analytic signal $f_a$ is then defined as $$ f_a(t) = 2 \int_{0}^\infty \hat{f}(x) e^{2\pi i xt} dx\,. $$ One easily verifies that $\text{Re} f_a = f \in L^1(\mathbb{R})$, but is it true that $f_a \in L^1(\mathbb{R})$ itself?