I came across a property of the Fourier transform (as shown in the picture), and I am having trouble finding what theorem or identity it is. I have tried searching various source but had no luck. Maybe I just couldn't recognize it in another form. Could someone point me in the right direction? Thank you.
after repeated use of the Fundamental Fourier identity $$\frac1{2\pi}\int_{-\infty}^\infty\int_{-\infty}^\infty\,dk\,dx\,e^{i(k+k')x}\tilde\psi(k')\;=\;\tilde\psi(-k)\tag{3.27.15}$$
(Geophysical Fluid Dynamics by Pedlosky, second edition)${{}}$
This formula is a direct consequence of the Fourier inversion theorem.
Let me be more precise. The Fourier Transform is defined by the formula $\mathcal F(f) (\xi)= \int_{-\infty}^\infty f(x)e^{-ix\xi}dx$ and its inverse is given by $\mathcal F^{-1}(g)(x) = \frac{1}{2\pi} \int_{-\infty}^\infty g(\xi) e^{ix\xi} d\xi$.
The Fourier inversion theorem says $\mathcal F \circ \mathcal F^{-1} = \mathcal F^{-1} \circ \mathcal F = \textrm{id}$.
The identity you want is $\mathcal F \circ \mathcal F^{-1}(\tilde\psi)(-k) = \tilde \psi(-k)$. Indeed \begin{align*}\tilde \psi(-k) &= \mathcal F\circ \mathcal F^{-1}(\tilde\psi)(-k) \\&=\int_{-\infty}^\infty \frac{1}{2\pi}\left(\int_{-\infty}^\infty \tilde \psi(x)e^{ix\xi}dx\right) e^{-ix (-k)}d\xi \\&= \frac{1}{2\pi}\int_{-\infty}^\infty\int_{-\infty}^\infty \tilde \psi(x)e^{ix (\xi+k)}dxd\xi. \end{align*}