Schwartz Function

Definition

We define the Schwartz functions $\mathcal{S}({\mathbb{R}}^n)$ to be the space of functions $f$ such that $f\in C^{\infty }({\mathbb{R}}^n)$ and

\lvert \partial^\alpha f(x) \rvert \lesssim \frac{C_{\alpha,N}}{(1 + \lvert x \rvert)^N},$$

where $\alpha$ is taking all possible derivatives and all $N\in \mathbb{N}$. If a function is Schwartz, then we may also call it rapidly decreasing. Or equivalently we have the following definition for Schwarz Space.

Schwarz Space

$\mathcal{S}({\mathbb{R}}^n)=\left\{f\in C^{\infty }({\mathbb{R}}^n):\forall \alpha ,\beta \in {\mathbb{N}}_0^n,{\sup }_{x\in {\mathbb{R}}^n}|x^{\alpha }{D}^{\beta }f(x)|<\infty \right\}$ $D^{\beta }f(x)$ represents the partial derivative of $f$ of order $|\beta |$: $D^{\beta }f=\frac{{\partial }^{|\beta |}f}{\partial x_1^{{\beta }_1}\partial x_2^{{\beta }_2}\cdots \partial x_n^{{\beta }_n}}$ So Schwarz space is just the space of infinitely differentiable functions that decrease rapidly at infinity, along with all their derivatives.

Proposition

Schwarz Space is dense in $L^p({\mathbb{R}}^n)$ for $1\le p<\infty$. It follows from $C_c^{\infty }({\mathbb{R}}^n)$ is dense in $L^p({\mathbb{R}}^n)$ and $C_c^{\infty }({\mathbb{R}}^{n)}\subset \mathcal{S}({\mathbb{R}}^n)$.