Fourier inversion

If the Fourier transform records how much of each frequency a signal contains, Fourier inversion plays the recording back: it reassembles the original signal by adding up all its frequency components, each with the right amplitude and phase. The remarkable point is that no information was lost — the function and its spectrum determine each other.

With the convention f-hat(xi) = integral of f(x) e^{-2*pi*i*x*xi} dx, the inversion formula is f(x) = integral of f-hat(xi) e^{+2*pi*i*x*xi} dxi. The only change is the sign in the exponent. But this clean statement needs hypotheses: it holds verbatim, at every point, when both f and f-hat are integrable (and f is continuous). When the integral does not converge absolutely, one interprets it as a symmetric limit or invokes summability, much as with Fourier series.

There is a subtlety worth flagging. At a point where f has a jump, the inversion integral (read as a principal value) returns the average (f(x+) + f(x-))/2, not either one-sided value — the exact analogue of Dirichlet's theorem for series. And inversion in the L2 sense (via Plancherel) recovers f only as an element of L2, that is up to a set of measure zero, so ‘the same function’ means almost everywhere equal.