Can you explain in which sense, for a generalized function $$ Y^+(\phi):=\int_0^\infty e^{i\phi t}dt$$ we have $(-i\phi)Y^+(\phi)=1$ ?
Here $\phi \in S^1$.
I clearly get that $\int_0^\infty e^{i\phi t}dt= \frac{1}{i\phi}\left.(e^{i\phi t})\right|_0^\infty$ and distribution means some nice behavior at infinity, but I'm kind of missig the details