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A unitary linear operator which resolves a function on $\mathbb{R}^N$ into a linear superposition of "plane wave functions". Most often used in physics for calculating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a quantum state in position co-ordinates into one in momentum co-ordinates and contrawise. There is also a discrete, fast Fourier transform for discretised data.

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What are the units of a convolution?

The only thing in the integral that has units, besides the integrand, is the measure $\mathrm{d}\tau$. So whatever the dimensions of $f$ and $g$ are, the convolution has dimensions $[f][g][t]$, since …
Wojciech Morawiec's user avatar