“Harish-Chandra's Schwartz space”的意思、由来-开放百科全书

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Harish-Chandra's Schwartz space

释义

Definition
References

In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by {{harvs|txt|last=Harish-Chandra|authorlink=Harish-Chandra|year=1966|loc=section 9}}. It is an analogue of the Schwartz space on a real vector space, and is used to define the space of tempered distributions on a semisimple Lie group.

Definition

The definition of the Schwartz space uses Harish-Chandra's Ξ function and his σ function. The σ function is defined by

for x=k exp X with k in K and X in p for a Cartan decomposition G = K exp p of the Lie group G, where ||X|| is a K-invariant Euclidean norm on p, usually chosen to be the Killing form. {{harv|Harish-Chandra|1966|loc=section 7}}.

The Schwartz space on G consists roughly of the functions all of whose derivatives are rapidly decreasing compared to Ξ. More precisely, if G is connected then the Schwartz space consists of all smooth functions f on G such that

is bounded, where D is a product of left-invariant and right-invariant differential operators on G {{harv|Harish-Chandra|1966|loc=section 9}}.

References

{{Citation | last1=Harish-Chandra | title=Discrete series for semisimple Lie groups. II. Explicit determination of the characters | doi=10.1007/BF02392813 |mr=0219666 | year=1966 | journal=Acta Mathematica | issn=0001-5962 | volume=116 | pages=1–111}}
{{Citation | last1=Wallach | first1=Nolan R | title=Real reductive groups. I | publisher=Academic Press | location=Boston, MA | series=Pure and Applied Mathematics | isbn=978-0-12-732960-4 |mr=929683 | year=1988 | volume=132}}

2 : Harmonic analysis|Representation theory

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