convolution theorem

Take a sharp photo and a description of how a lens blurs a point of light. The blurred photo is built by smearing each point of the sharp image according to that blur, then adding the smears up. This smearing-and-summing operation is called convolution, and it is everywhere: it describes blurring, echo, smoothing, the output of any linear time-invariant system. The convolution theorem is the single fact that makes convolution tractable — it turns this awkward sliding-and-overlapping operation into ordinary pointwise multiplication.

Convolution of two functions is (f star g)(x) = integral of f(t) g(x minus t) dt: you flip g, slide it past f, and at each offset record the area of their overlap. The theorem states that the Fourier transform of a convolution is the product of the transforms: FT{f star g} = F(k) G(k). And dually, the transform of a product is (a scaled) convolution of the transforms. So filtering — which in the time domain is a convolution of the signal with the filter's impulse response — becomes, in the frequency domain, just multiplying the signal's spectrum by the filter's frequency response, frequency by frequency.

This is why engineers and physicists live in the frequency domain. A low-pass filter is described by a frequency response that is near 1 at low frequencies and near 0 at high ones; applying it is multiplication there, trivially understood, whereas the equivalent time-domain convolution is an integral you would rather not compute by hand. The theorem underlies image deblurring, audio equalization, the solution of differential equations with Green's functions (a Green's function acts by convolution), and — combined with the fast Fourier transform — the fast multiplication of large numbers and polynomials. The honest caveat is that the clean product rule assumes the transforms exist and the convolution integral converges.