Is there anything special one can say about the fourier transform
$$\hat g(y)=\frac{1}{2\pi}\int_{-\infty}^\infty dx e^{i x y} g(x)$$
of the function
$$g(x)=\theta(x-L)f(x)$$
where $L>0$, $\theta(x-L)$ is the Heaviside step function and $f(x)$ is some well behaved test function?