Schwartz space

Let \mathbb{N} be the set of non-negative integers, and for any {{tmath| n \in \mathbb{N} }}, let \mathbb{N}^n := \underbrace{\mathbb{N} \times \dots \times \mathbb{N}}_{n \text{ times}} be the -fold Cartesian product. The Schwartz space or space of rapidly decreasing functions on \mathbb{R}^n is the function space\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}} where C^{\infty}(\mathbb{R}^n, \mathbb{C}) is the function space of smooth functions from \mathbb{R}^n into {{tmath| \mathbb{C} }}, and \|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, \sup denotes the supremum, and we used multi-index notation, i.e. \boldsymbol{x}^\boldsymbol{\alpha} := x_1^{\alpha_1}x_2^{\alpha_2}\ldots x_n^{\alpha_n} and {{tmath|1= 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 f such that , , {{tmath| f^{\prime\prime}(x) }}, ... all exist everywhere on \mathbb{R} and go to zero as x \rightarrow \pm \infty faster than any reciprocal power of . In particular, \mathcal{S}\left(\mathbb{R}^n, \mathbb{C}\right) is a subspace of {{tmath| C^{\infty}(\mathbb{R}^n, \mathbb{C}) }}. == Examples of functions in the Schwartz space ==

• If \boldsymbol{\alpha} is a multi-index, and a is a positive real number, then {{tmath| \boldsymbol{x}^\boldsymbol{\alpha} e^{-a \vert \boldsymbol{x} \vert^2} \in \mathcal{S}(\mathbb{R}^n) }}. • Any smooth function f with compact support is in {{tmath| \mathcal{S}\left(\mathbb{R}^n\right) }}. This is clear since any derivative of f is continuous and supported in the support of , so (\boldsymbol{x}^\boldsymbol{\alpha}\boldsymbol{D}^\boldsymbol{\alpha})f has a maximum in \mathbb{R}^n by the extreme value theorem. • Because the Schwartz space is a vector space, any polynomial \phi(\boldsymbol{x}) can be multiplied by a factor e^{-a \vert\boldsymbol{x}\vert^2} for 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. == Properties ==

Analytic properties • From Leibniz's rule, it follows that \mathcal{S}\left(\mathbb{R}^n\right) is also closed under pointwise multiplication: {{tmath| f, g \in \mathcal{S}\left(\mathbb{R}^n\right) }} implies {{tmath| fg \in \mathcal{S}\left(\mathbb{R}^n\right) }}. In particular, this implies that \mathcal{S}\left(\mathbb{R}^n\right) is an \mathbb{R}-algebra. More generally, if f \in\mathcal{S}\left(\mathbb{R}\right) and H is a bounded smooth function with bounded derivatives of all orders, then {{tmath| fH \in\mathcal{S}\left(\mathbb{R}\right) }}. • The Fourier transform is a linear isomorphism {{tmath| \mathcal F:\mathcal{S}\left(\mathbb{R}^n\right) \rightarrow \mathcal{S}\left(\mathbb{R}^n\right) }}. • If f \in \mathcal{S}\left(\mathbb{R}^n\right) then f is Lipschitz continuous and hence uniformly continuous on {{tmath| \mathbb{R}^n }}. • \mathcal{S}\left(\mathbb{R}^n\right) is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers. • Both \mathcal{S}\left(\mathbb{R}^n\right) and its strong dual space are also: • complete Hausdorff locally convex spaces, • nuclear Montel spaces, • ultrabornological spaces, • reflexive barrelled Mackey spaces. Relation of Schwartz spaces with other topological vector spaces • If , then \mathcal{S}\left(\mathbb{R}^n\right) is a dense subset of {{tmath| L^p(\mathbb{R}^n) }}. • The space of all bump functions, {{tmath| C_\text{c}^\infty\left(\mathbb{R}^n\right) }}, is included in {{tmath| \mathcal{S}\left(\mathbb{R}^n\right) }}. == See also ==