Suppose real function g(t) has corresponding fourier transform G(f). In one text book I saw that the complex conjugate of G(f) equals G(-f). How to prove this?
ie for a real valued function $g(t)$, how to prove that $G^{*}(f) = G(-f)$, where $G(f)$ is the fourier transform of $g(t)$.
Any help or links to online references will be very much appreciated.
It is straightfoward to prove from the definition of the Fourier transform: \begin{align} G^*(f) &= \int \overline{g(t) e^{-2\pi i f t}} \; dt \\ &= \int \overline{g(t)} e^{2\pi i f t} \; dt \\ &= \int g(t) e^{-2\pi i (-f) t} \; dt \\ &= G(-f). \end{align} Note that proving the basic properties of the Fourier transform and convolution boils down to manipulating integrals (usually a subtitution or Fubini's theorem does the trick).
