Let $f, g \in L^1(\mathbb{R})$ be two functions that satisfy the following conditions:
$(1)$ $f(\mathbb{R}), g(\mathbb{R}) \subseteq \mathbb{R}$,
$(2)$ $||f||_{L^1} \leq 1$, $||g||_{L^1} \leq 1$,
$(3)$ $f(-x) = -f(x)$ and $g(-x)=-g(x)$ for all $x \in \mathbb{R}$
$(4)$ $|\hat{f} + 1 | = |\hat{g} +1|$, where $\hat{f}, \hat{g}$ deonotes the Fourier transforms.
Can we conclude from this that $f=g$? If not, what counter examples are there?