Inverse Fourier transform

Summary

The inverse Fourier transform takes a function of frequency and builds a function of position by superposing the waves $e^{2\pi i x \cdot \xi}$ with these amplitudes. It has the same form as the Fourier transform, with the opposite sign in the exponent. The Fourier inversion theorem says that it undoes the Fourier transform, so a function can be recovered from its frequency content.