Fourier transform of $e^{-x} H(x)$

Question

Find the Fourier Transform of $e^{-x}H(x)$.

I can find the answer using general definition of Fourier transform. But I find it in the exercise Distribution chapter. So, I want to do it using distribution theory.

I have written ${f}\hat [\delta] = f[\delta^\hat\ ] = \int H(x) \phi^\hat \ (x) dx$.

But I can not proceed further. Please help me.

Answer 1

If you insist on using facts from distribution theory you could start with the fact that $f(x)= e^{-x} H(x)$ satisfies the distribution identity $f'(x)+f(x)= e^{-x} H'=\delta$