Fourier Transform FT

Question

Two questions:

• Are there more than one definitions of the Fourier transform?
• Called $g(\omega)$ the FT of a real function $f(t)$, is $g(\omega) = g^{*}(- \omega)$?

Thanks

Answer 1

Yes there are different definitions in different fields of application. Some choices are $$F(s) =\mathcal{F}\{f(t)\} = \frac{1}{A}\int_{-\infty}^{+\infty}e^{iBst}f(t)dt$$ Where $$A = \sqrt{2\pi} , B = \pm1\\A = 1 , B = \pm2\pi \\ A = 1 , B = \pm1$$For complete table refer to Fourier Analysis by T. W. Körner. If $A = 1$ and $B = -2\pi$ then using this we have $$\mathcal{F}\{f^*(t)\} = \int_{-\infty}^{+\infty}e^{-i2\pi st}f^{*}(t)dt = (\int_{-\infty}^{+\infty}e^{i2\pi st}f(t)dt)^* =F^{*}(-s)$$ Note that the order of two operations i.e. flipping the frequency variable and conjugating doesn't matter here and the result is the same. In particular if $f(t)$ is real then $F(s) = F^*(-s)$ which means $F^*(s) = F(-s)$. So in the real case, for knowing $F(s)$ we need only $F(s)$ for $s\ge 0$ or $s \le 0$.