The Schwartz space or space of rapidly decreasing functions on \(\mathbb{R}^d\) is the function space
\[ \mathcal{S}(\mathbb{R}^d,\mathbb{C})=\{f\in C^\infty(\mathbb{R}^d,\mathbb{C})\mid \forall \alpha,\beta\in \mathbb{N}^d\colon \lVert f\rVert_{\alpha,\beta}<\infty\}, \]where \(C^\infty(\mathbb{R}^d,\mathbb{C})\) is the space of smooth functions from \(\mathbb{R}^d\) to \(\mathbb{C}\), and
\[ \lVert f\rVert_{\alpha,\beta}=\sup_{x\in \mathbb{R}^d}\lvert x^\alpha(\partial^\beta f)(x)\rvert. \]
Remarks
- In essence the Schwartz space is the space of all smooth functions all derivatives of which decay faster at infinity than any polynomial.
- \(\mathcal{S}(\mathbb{R}^n)\subseteq L^p(\mathbb{R}^n)\)