Schwartz space

Let ${\displaystyle \mathbb {N} }$ be the set of non-negative integers, and for any ${\displaystyle n\in \mathbb {N} }$, let ${\displaystyle \mathbb {N} ^{n}:=\underbrace {\mathbb {N} \times \dots \times \mathbb {N} } _{n{\text{ times}}}}$ be the n-fold Cartesian product. The Schwartz space or space of rapidly decreasing functions on ${\displaystyle \mathbb {R} ^{n}}$ is the function space

$${\displaystyle S\left(\mathbb {R} ^{n},\mathbb {C} \right):=\left\{f\in C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )\mid \forall \alpha ,\beta \in \mathbb {N} ^{n},\|f\|_{\alpha ,\beta }<\infty \right\},}$$

where ${\displaystyle C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )}$ is the function space of smooth functions from ${\displaystyle \mathbb {R} ^{n}}$ into ${\displaystyle \mathbb {C} }$, and

$${\displaystyle \|f\|_{\alpha ,\beta }:=\sup _{x\in \mathbb {R} ^{n}}\left|x^{\alpha }(D^{\beta }f)(x)\right|.}$$

Here, ${\displaystyle \sup }$ denotes the supremum, and we used multi-index notation, i.e. ${\displaystyle x^{\alpha }:=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}}}$ and ${\displaystyle D^{\beta }:=\partial _{1}^{\beta _{1}}\partial _{2}^{\beta _{2}}\ldots \partial _{n}^{\beta _{n}}}$.

To put common language to this definition, one could consider a rapidly decreasing function as essentially a function f(x) such that f(x), f ′(x), f ′′(x), ... all exist everywhere on R and go to zero as x→ ±∞ faster than any reciprocal power of x. In particular, S(Rⁿ, C) is a subspace of the function space C^∞(Rⁿ, C) of smooth functions from Rⁿ into C.

• If α is a multi-index, and a is a positive real number, then

  ${\displaystyle x^{\alpha }e^{-a|x|^{2}}\in {\mathcal {S}}(\mathbf {R} ^{n}).}$

• Any smooth function f with compact support is in S(Rⁿ). This is clear since any derivative of f is continuous and supported in the support of f, so (x^αD^β) f has a maximum in Rⁿ by the extreme value theorem.
• Because the Schwartz space is a vector space, any polynomial ${\displaystyle \phi (x^{\alpha })}$ can by multiplied by a factor ${\displaystyle e^{-ax^{2}}}$ for ${\displaystyle a>0}$ a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials inside a Schwartz space.

• Bump function
• Schwartz–Bruhat function
• Nuclear space