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$${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n},\mathbb {C} \right):=\left\{f\in C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )\mid \forall {\boldsymbol {\alpha }},{\boldsymbol {\beta }}\in \mathbb {N} ^{n},\|f\|_{{\boldsymbol {\alpha }},{\boldsymbol {\beta }}}<\infty \right\},}$$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$${\displaystyle \|f\|_{{\boldsymbol {\alpha }},{\boldsymbol {\beta }}}:=\sup _{{\boldsymbol {x}}\in \mathbb {R} ^{n}}\left|{\boldsymbol {x}}^{\boldsymbol {\alpha }}({\boldsymbol {D}}^{\boldsymbol {\beta }}f)({\boldsymbol {x}})\right|.}$$ Here, ${\displaystyle \sup }$ denotes the supremum, and we used multi-index notation, i.e. ${\displaystyle {\boldsymbol {x}}^{\boldsymbol {\alpha }}:=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}}}$ and ${\displaystyle D^{\boldsymbol {\beta }}:=\partial _{1}^{\beta _{1}}\partial _{2}^{\beta _{2}}\ldots \partial _{n}^{\beta _{n}}}$.

To put common language to this definition, one could consider a rapidly decreasing function as essentially a function ${\displaystyle f}$ such that ${\displaystyle f(x),f'(x),f^{\prime \prime }(x),\ldots }$ , all exist everywhere on ${\displaystyle \mathbb {R} }$ and go to zero as ${\displaystyle x\rightarrow \pm \infty }$ faster than any reciprocal power of ${\displaystyle x}$. In particular, ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n},\mathbb {C} \right)}$ is a subspace of ${\displaystyle C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )}$.

• If ${\displaystyle {\boldsymbol {\alpha }}}$ is a multi-index, and a is a positive real number, then

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

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

• From Leibniz's rule, it follows that ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ is also closed under pointwise multiplication:

${\displaystyle f,g\in {\mathcal {S}}\left(\mathbb {R} ^{n}\right)\Rightarrow fg\in {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ In particular, this implies that ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ is an ${\displaystyle \mathbb {R} }$-algebra. More generally, if ${\displaystyle f\in {\mathcal {S}}\left(\mathbb {R} \right)}$ and ${\displaystyle H}$ is a bounded smooth function with bounded derivatives of all orders, then ${\displaystyle fH\in {\mathcal {S}}\left(\mathbb {R} \right)}$.

• The Fourier transform is a linear isomorphism ${\displaystyle {\mathcal {F}}:{\mathcal {S}}\left(\mathbb {R} ^{n}\right)\longrightarrow {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$.
• If ${\displaystyle f\in {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ then ${\displaystyle f}$ is Lipschitz continuous and hence uniformly continuous on ${\displaystyle \mathbb {R} ^{n}}$.
• ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers.
• Both ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ and its strong dual space are also:

1. complete Hausdorff locally convex spaces,
2. nuclear Montel spaces,
3. ultrabornological spaces,
4. reflexive barrelled Mackey spaces.

• If ${\displaystyle 1\leqslant p<\infty }$, then ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ is a dense subset of ${\displaystyle L^{p}(\mathbb {R} ^{n})}$.
• The space of all bump functions, ${\displaystyle C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)}$, is included in ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$.

• Bump function
• Schwartz–Bruhat function
• Nuclear space