Questions tagged [schwartz-space]

The Schwartz space on $\mathbb R^n$, usually denoted $\mathcal S(\mathbb R^n)$ or just $\mathcal S$ for short is the vector space of all smooth functions $f:\mathbb R^n\to\mathbb R$ such that for all multi-indices $\alpha,\beta\in(\mathbb N\cup\{0\})^n$, the following semi-norms are finite:

$$ |f|_ {\alpha,\beta} := \sup _{x\in\mathbb R^n} |x^\alpha \partial^\beta f(x)| <\infty.$$

Here, $x^\alpha := x_1^{\alpha_1} \dots x_n ^{\alpha_n}$ and $\partial^\beta := \frac{\partial^{|\beta|}}{\partial x_1^{\beta_1}\dots \partial x_n^{\beta_n}}$. Intuitively, $f$ and all its derivatives decay faster than any polynomial.

Questions about Schwartz functions naturally arise when discussing distribution-theory, fourier-analysis, fractional-sobolev-spaces, or harmonic-analysis, which are topics in real-analysis.

Every compactly supported smooth function is Schwartz (i.e., $\mathcal D \subset \mathcal S$), and $e^{-|x|^2}\in\mathcal S$ is not compactly supported. The semi-norms generate a topology that is metrizable but not normable.

One reason why $\mathcal S$ is important is that the Fourier Transform of a function in $\mathcal S$ is easily seen to also be in $\mathcal S$. The boundedness of the seminorms mean that many formal manipulations like differentiating under the integral sign are very easy to justify when using Schwartz functions. For example, one defines a Fourier inversion first on $\mathcal S$, which extends to $L^2$. More generally, Fourier multipliers are defined by multiplication 'on the Fourier side', at least when the functions are in $\mathcal S$.

The dual of $\mathcal S$ is called the space of tempered distributions. As mentioned $\mathcal D \subset \mathcal S$, and therefore $\mathcal D' \supset \mathcal S'$. The distributions $T$ that are also tempered have a Fourier transform defined by duality, $$ \langle \mathcal FT, \phi\rangle := \langle T, \mathcal F\phi\rangle.$$ For example, $\mathcal F\delta_0 = 1$ in the sense of tempered distributions.

The Schwartz space can be similarly defined for other sets; for example the Schwartz space on the circle $\mathbb R/\mathbb Z$ is coincident with the space of smooth functions.

Wikipedia link here.