Schwartz space

Summary

The Schwartz space $\mathcal{S}(\R^n)$ is the space of smooth functions on $\R^n$ that, together with all their derivatives, decay faster than any power of $1/\abs{x}$. The Fourier transform maps it bijectively onto itself, which makes it the natural home of Fourier analysis on $\R^n$. Its dual space is the space of tempered distributions. See the definition.