Schwartz room
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.
The vector space of all Schwartz functions is called Schwartz space and is denoted by. In a nutshell, then
The Schwartz space is a locally convex space that can be metrised , which is defined by the family of semi-norms
is induced.
Examples
- The functions are based on Schwartz functions .
- 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.
- 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 .
- 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 .
- for .
- For the Schwartz space lies close to the space of the p-integrable functions
- With the help of this density argument one can define the Fourier transformation on the Hilbert space .
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 .
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 .