Quantization and Arithmetic (Paperback, 2008 ed.)

(12) (4) Let ? be the unique even non-trivial Dirichlet character mod 12, and let ? be the unique (odd) non-trivial Dirichlet character mod 4. Consider on the line the distributions m (12) ? d (x)= ? (m)? x? , even 12 m?Z m (4) d (x)= ? (m)? x? . (1.1) odd 2 m?Z 2 i?x UnderaFouriertransformation,orundermultiplicationbythefunctionx ? e , the?rst(resp. second)ofthesedistributionsonlyundergoesmultiplicationbysome 24th (resp. 8th) root of unity. Then, consider the metaplectic representation Met, 2 a unitary representation in L (R) of the metaplectic group G, the twofold cover of the group G = SL(2,R), the de?nition of which will be recalled in Section 2: it extends as a representation in the spaceS (R) of tempered distributions. From what has just been said, if g ~ is a point of G lying above g? G,andif d = d even g ~ ?1 or d , the distribution d =Met(g~ )d only depends on the class of g in the odd homogeneousspace?\G=SL(2,Z)\G,uptomultiplicationbysomephasefactor, by which we mean any complex number of absolute value 1 depending only on g ~. On the other hand, a function u?S(R) is perfectly characterized by its scalar g ~ productsagainstthedistributionsd ,sinceonehasforsomeappropriateconstants C , C the identities 0 1 g ~ 2 2 | d ,u | dg = C u if u is even, 2 0 even L (R) ?\G

General

Imprint:	Birkhauser Verlag AG
Country of origin:	Switzerland
Series:	Pseudo-Differential Operators, 1
Release date:	June 2008
First published:	2008
Authors:	Andr e Unterberger
Dimensions:	240 x 170 x 8mm (L x W x T)
Format:	Paperback
Pages:	147
Edition:	2008 ed.
ISBN-13:	978-3-7643-8790-7
Categories:	Books > Science & Mathematics > Mathematics > General
LSN:	3-7643-8790-4
Barcode:	9783764387907

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