the Fourier transform of a distribution

The Fourier transform rewrites a signal as a recipe of frequencies, telling you how much of each pure wave it contains. For ordinary nice functions this is a familiar integral. But many objects of PDE theory — a point source, a constant, a single plane wave — are not nice enough for that integral to converge. The Fourier transform of a distribution extends the transform to all such objects, so that frequency-space methods become fully rigorous.

The extension uses the same trick as distributional differentiation: move the hard operation onto the smooth probe. The Schwartz space is preserved by the Fourier transform, so for a tempered distribution T we define its transform by the rule (transform of T, phi) = (T, transform of phi). The probe phi is Schwartz, so its transform is again Schwartz and the right-hand pairing makes sense — and that single rule defines the transform of T for every probe at once. The two showcase results fall straight out: the transform of the delta is the constant 1 (a point in space is all frequencies in equal measure), and the transform of the constant 1 is the delta (a single frequency, namely zero, for something that never varies). Differentiation becomes multiplication by the frequency variable, and convolution becomes ordinary multiplication.

This matters because it is the clean home for transform methods on PDEs. Applying the Fourier transform turns a constant-coefficient PDE into an algebraic equation in frequency space — derivatives become multiplications — which you solve and transform back. Dispersion relations, the symbol of an operator, and frequency-space estimates all live here, now on a rigorous footing rather than a formal one.