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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Schwartz space \mathcal is the
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
of all functions whose
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space \mathcal^* of \mathcal, that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function. Schwartz space is named after French mathematician Laurent Schwartz.


Definition

Let \mathbb be the set of non-negative
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, and for any n \in \mathbb, let \mathbb^n := \underbrace_ be the ''n''-fold Cartesian product. The ''Schwartz space'' or space of rapidly decreasing functions on \mathbb^n is the function space\mathcal \left(\mathbb^n, \mathbb\right) := \left \,where C^(\mathbb^n, \mathbb) is the function space of smooth functions from \mathbb^n into \mathbb, and\, f\, _:= \sup_ \left, \boldsymbol^\boldsymbol (\boldsymbol^\boldsymbol f)(\boldsymbol) \. Here, \sup denotes the supremum, and we used multi-index notation, i.e. \boldsymbol^\boldsymbol:=x_1^x_2^\ldots x_n^ and D^\boldsymbol:=\partial_1^\partial_2^\ldots \partial_n^. To put common language to this definition, one could consider a rapidly decreasing function as essentially a function such that , , , ... all exist everywhere on and go to zero as faster than any reciprocal power of . In particular, is a subspace of the function space (, ) of smooth functions from into .


Examples of functions in the Schwartz space

* If \boldsymbol is a multi-index, and ''a'' is a positive
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
, then *:\boldsymbol^\boldsymbol e^ \in \mathcal(\mathbb^n). * Any smooth function ''f'' with
compact support In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed ...
is in . This is clear since any derivative of ''f'' is continuous and supported in the support of ''f'', so (\boldsymbol^\boldsymbol\boldsymbol^\boldsymbol)f has a maximum in R''n'' by the extreme value theorem. * Because the Schwartz space is a vector space, any polynomial \phi(\boldsymbol) can be multiplied by a factor e^ 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 is also closed under pointwise multiplication: *: If then the product . In particular, this implies that is an -algebra. More generally, if and is a bounded smooth function with bounded derivatives of all orders, then . * The Fourier transform is a linear isomorphism . * If then is Lipschitz continuous and hence uniformly continuous on . * is a distinguished locally convex Fréchet Schwartz TVS over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. * Both ''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 . *If , then is dense in . *The space of all
bump function In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
s, , is included in .


See also

*
Bump function In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
* Schwartz–Bruhat function * Nuclear space


References


Sources

* * * * {{Functional analysis Topological vector spaces Smooth functions Fourier analysis Function spaces Schwartz distributions