I am looking for a rule to get the Fourier transform of $f(t)u(t)$ where $u(t)$ is Heaviside step function. In other words, assume I know $\mathcal Ff$ over the full real line $-\infty < t < \infty$, what is the Fourier transform of f restricted to $t>0$? More generally, is there a rule for the Fourier transform of $f$ restricted over a finite interval $a < t < b$?
The usual Fourier transform tables found online don't have many functional relationship rules. Any good reference to more detailed tables would be very helpful!
My attempt: $\mathcal F[f\times u] = (\mathcal Ff)*(\mathcal Fu)$ where * denotes convolution. From $\mathcal Fu(\lambda) = -\frac i\lambda + \pi\delta(\lambda)$ we get
$$ \mathcal F[f\times u](\lambda) = -i \left[\mathcal Ff*(\frac1t)\right](\lambda) + \pi\mathcal Ff(\lambda) = -i \int_{-\infty}^\infty \frac{f(t)}{\lambda - t} dt + \pi\mathcal Ff(\lambda) = -i\mathcal Hf(\lambda) + \pi\mathcal Ff(\lambda) $$ where $\mathcal H$ denotes the Hilbert transform.