Schwartz room

Graph of the two-dimensional Gaussian bell curve

The Schwartz space is a function space that is examined in the mathematical sub-area of functional analysis. This is named after the mathematician Laurent Schwartz , who provided central results in distribution theory , whereby the Schwartz space also played an important role. The elements of the Schwartz space are called Schwartz functions . A special feature of this space is that the Fourier transform forms a linear automorphism in this space.

definition

A function is called a Schwartz function or rapidly decreasing if it is continuously differentiable any number of times , and if the function is restricted to for all multi-indices , where the -th derivative denotes. ${\ displaystyle f \ colon \ mathbb {R} ^ {n} \ rightarrow \ mathbb {C}}$ ${\ displaystyle \ alpha, \ beta \ in \ mathbb {N} _ {0} ^ {n}}$ ${\ displaystyle x ^ {\ alpha} D ^ {\ beta} f (x)}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle D ^ {\ beta}}$ ${\ displaystyle \ beta}$

The vector space of all Schwartz functions is called Schwartz space and is denoted by. In a nutshell, then ${\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n})}$

{\ displaystyle {\ begin {aligned} {\ mathcal {S}} (\ mathbb {R} ^ {n}) \; & {\ overset {} {: =}} \; \ left \ {\ phi \ in C ^ {\ infty} (\ mathbb {R} ^ {n}) \, {\ Big |} \, \ forall \ alpha, \ beta \ in \ mathbb {N} _ {0} ^ {n}: \ ; \ sup _ {x \ in \ mathbb {R} ^ {n}} | x ^ {\ alpha} D ^ {\ beta} \ phi (x) | <\ infty \; \ right \} \\ [. 4em] & = \ left \ {\ phi \ in C ^ {\ infty} (\ mathbb {R} ^ {n}) \, {\ Big |} \, \ forall \ alpha, \ beta \ in \ mathbb { N} _ {0} ^ {n}, \, \ exists C \ geq 0, \, \ forall x \ in \ mathbb {R} ^ {n}: \; | x ^ {\ alpha} D ^ {\ beta} \ phi (x) | \ leq C \; \ right \} \,. \ end {aligned}}}

The Schwartz space is a locally convex space that can be metrised , which is defined by the family of semi-norms

{\ displaystyle \ | f \ | _ {N} = \ sup _ {x \ in \ mathbb {R} ^ {n}} \ max _ {| \ alpha |, \, | \ beta | <N} | x ^ {\ alpha} D ^ {\ beta} f (x) |}

is induced.

Examples

The functions are based on Schwartz functions . ${\ displaystyle \ exp (-a \ | x \ | ^ {2})}$ ${\ displaystyle a> 0}$ ${\ displaystyle \ mathbb {R} ^ {n}}$
Every function that can be differentiated as often as required with a compact carrier is a Schwartz function. The vector space of the test functions with compact support is therefore a real subspace of the Schwartz space. ${\ displaystyle C_ {c} ^ {\ infty} (\ mathbb {R} ^ {n}) \ cong {\ mathcal {D}} (\ mathbb {R} ^ {n})}$
The Hermitian functions are also Schwartz functions.

properties

The Schwartz space is complete in terms of the topology (or metric) induced by the family of semi-norms , and is thus a Fréchet space . He also has the Montel property .

{\ displaystyle (\ | \ cdot \ | _ {N}) _ {N}}

The Fourier transform forms a linear automorphism on Schwartz space.

As mentioned in the examples, the space of smooth functions with compact support is a subspace of the Schwartz space. This is even close to the Schwartz room.

The Schwartz room is separable .

{\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n}) \ subset L ^ {p} (\ mathbb {R} ^ {n})}

for .

{\ displaystyle 1 \ leq p \ leq \ infty}

For the Schwartz space lies close to the space of the p-integrable functions ${\ displaystyle 1 \ leq p <\ infty}$ ${\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n})}$ ${\ displaystyle L ^ {p} (\ mathbb {R} ^ {n})}$
With the help of this density argument one can define the Fourier transformation on the Hilbert space . ${\ displaystyle L ^ {2} (\ mathbb {R} ^ {n})}$

Tempered distributions

A continuous , linear mapping is called temperature-controlled distribution . The set of all tempered distributions is denoted by. This is the topological dual space too . ${\ displaystyle f \ colon {\ mathcal {S}} (\ mathbb {R} ^ {n}) \ rightarrow \ mathbb {C}}$ ${\ displaystyle {\ mathcal {S}} '(\ mathbb {R} ^ {n})}$ ${\ displaystyle {\ mathcal {S}} (\ mathbb {R} ^ {n})}$

literature

Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).

Individual evidence

↑ ^a ^b Man Wah Wong: An introduction to pseudo-differential operator . World Scientific, River Edge, NJ 1999, ISBN 978-981-02-3813-1 , pp. 10-11 .

[wong1011-1] Man Wah Wong: An introduction to pseudo-differential operator . World Scientific, River Edge, NJ 1999, ISBN 978-981-02-3813-1 , pp. 10-11 .