space of rapidly decreasing functions

The function space of rapidly decreasing functions $𝒮$ has theimportant property that the Fourier transform is an endomorphismon this space. This property enables one, by duality, todefine the Fourier transformfor elements in the dual space of $𝒮$, that is, for tempereddistributions.

DefinitionThe space of rapidly decreasing functions on ${ℝ}^n$is the function space

where $C^{\mathrm{\infty }}({ℝ}^n)$ is the set of smooth functionsfrom ${ℝ}^n$ to $ℂ$, and

$${||f||}_{\alpha ,\beta }={||x^{\alpha }{D}^{\beta }f||}_{\mathrm{\infty }}.$$

Here, $||\cdot |{|}_{\mathrm{\infty }}$ is the supremum norm, and we usemulti-index notation.When the dimension $n$ is clear, it is convenient to write$𝒮=𝒮({ℝ}^n)$. The space $𝒮$ is also called theSchwartz space, after Laurent Schwartz(1915-2002) [2].

0.0.1 Examples of functions in $𝒮$

1. If $i$ is a multi-index, and $a$ is a positivereal number, then

   $$x^i\exp \{-a{x}^2\}\in 𝒮.$$

2. Any smooth function with compact support $f$ is in $𝒮$.This is clear since any derivative of $f$ is continuous, so$x^{\alpha }{D}^{\beta }f$ has a maximum in ${ℝ}^n$.

0.0.2 Properties

1. $𝒮$ is a complex vector space. In other words,$𝒮$ is closed under point-wise addition and undermultiplication by a complex scalar.

2. Using Leibniz’ rule, it follows that $𝒮$ is also closedunder point-wise multiplication; if $f,g\in 𝒮$, then$fg:x\mapsto f(x)g(x)$ is also in $𝒮$.

3. For any $1\le p\le \mathrm{\infty }$, we have [3]

   $$𝒮\subset L^p,$$

   and if , then $𝒮$ is also dense in $L^p$.

4. The Fourier transform is a linear isomorphism $𝒮\to 𝒮$.