In his famous 1986 Dirac Memorial Lecture, Feynman refers to a Fourier transforms theorem holding in case F(w) satisfies "certain properties", while being restricted to positive frequencies only:

extract from lecture
as I am interested in better understanding said "certain properties", I have GOOGLEd around in search of a source for said theorem, but could not find a way to single it out from hundreds of search results (mostly related to DFT techniques). Would anybody know the exact naming of such theorem, and-or about any Fourier transforms properties publication describing it?

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    $\begingroup$ You would probably get a better answer on Mathematics.SE. This is a question about Fourier analysis, not physics. $\endgroup$ – knzhou Mar 5 '19 at 12:54
  • $\begingroup$ though it does have very important implications on the physics of particles as well. Feynman continues on with discussing non zero amplitudes for paths outside the light cone, anti-particles justification, etc. $\endgroup$ – Luca Mar 5 '19 at 12:59

His point is that if the integral defining $f(t)$ converges for all real $t$ then $f(t)$ on the real axis is the boundary value of a function that is analytic in the lower half plane. (Observe that taking $t\to t-i\tau$, $\tau>0$ improves the convergence of the integral) Now analytic functions that have a limit point of zeros (as is guaranteed by their vanishing in a finite interval) in the interior of their domain of analyticity have to vanish everywhere in that domain. What is not immediately clear to me to what extent this is true for functions vanishing the boundary of their domain as the boundary limits can be quite singular. For such limits, I suggest that you look up "Hardy Space" on Wikipedia.

  • $\begingroup$ I am not sure whether that is a proof or just a sketch of how yourself would proceed to prove such a theorem. In that case, I am afraid I am not proficient enough to complete that sketch on my own. But sure, somewhere there shall exist a rigorous proof precisely identifying said "certain properties". $\endgroup$ – Luca Mar 5 '19 at 14:49
  • $\begingroup$ @Luca Agreed. It's a sketch of what Feynman must have had in mind. What one needs is sufficient conditions to make the boundary value $f(t)$ analytic. Sufficiently fast vanishing of $F(\omega)$ as $\omega$ gets large should be enough. I'm pretty sure that if $|F(\omega)|$ is bounded by an exponential $e^{-\mu \omega}$, $\mu>0$ then $f(t)$ will be analytic on the real axis and then the claimed result holds. $\endgroup$ – mike stone Mar 5 '19 at 14:55
  • $\begingroup$ I just came across this book [Fourier Analysis, Self-Adjointness. Michael Reed, Barry Simon ](elsevier.com/books/ii-fourier-analysis-self-adjointness/reed/…) offering a free preview of page 19, which is indeed introducing the subject basically on the same lines as you sketched. Though running through the details of the various demonstrations appears quite demanding. $\endgroup$ – Luca Mar 6 '19 at 9:34

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