the Schwartz space
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The Schwartz space is a slightly larger family of probes than the compactly supported test functions, designed specifically so that the Fourier transform feels at home. A compactly supported probe is great for differentiation but the Fourier transform tends to spread its support out to infinity, so it can leave the family. The Schwartz space fixes this by allowing functions that stretch out to infinity, provided they die away faster than any polynomial can grow.
Precisely, a Schwartz function is an infinitely smooth function such that the function itself and every one of its derivatives decays faster than any inverse power of x as x goes to infinity — that is, x^n times any derivative stays bounded for every power n. A Gaussian like exp(-x^2) is the prototype: smooth, and its tails vanish faster than 1/x^n for every n. The decisive feature is balance under the Fourier transform: the transform of a Schwartz function is again a Schwartz function, and the transform turns differentiation into multiplication by the variable and vice versa, so smoothness and decay trade places but never leave the space. The Schwartz space is written S.
This space matters because it is the right launchpad for the Fourier transform of distributions. The distributions that pair continuously with Schwartz probes are the tempered distributions, and the Fourier transform extends to them cleanly precisely because S is preserved. Compactly supported test functions sit inside the Schwartz space, so every Schwartz-based statement also applies to ordinary test functions.
The Gaussian g(x) = exp(-x^2) is a Schwartz function. Differentiate it any number of times and you still get a smooth function whose tails vanish faster than any power; its Fourier transform is again a Gaussian, the cleanest possible demonstration that S maps to itself.
Smooth with super-polynomial decay — and the Fourier transform keeps you inside the family.
Compact support implies Schwartz decay, so every compactly supported test function is a Schwartz function, but not the reverse: the Gaussian is Schwartz yet never exactly zero, so it is not compactly supported.