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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Functions on noncompact Lie groups with positive Fourier transforms


Author: Takeshi Kawazoe
Journal: Proc. Amer. Math. Soc. 123 (1995), 1411-1415
MSC: Primary 42A38; Secondary 22E30, 43A30
MathSciNet review: 1277119
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Abstract: Let G be a homogeneous group with the graded Lie algebra or a noncompact semisimple Lie group with finite center. We define the Fourier transform $ \hat f$ of f as a family of operators $ \hat f(\pi ) = {\smallint _G}f(x)\pi (x)dx(\pi \in \hat G)$, and we say that $ \hat f$ is positive if all $ \hat f(\pi )$ are positive. Then, we construct an integrable function f on G with positive $ \hat f$ and the restriction of f to any ball centered at the origin of G is square-integrable, however, f is not square-integrable on G.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1277119-7
PII: S 0002-9939(1995)1277119-7
Article copyright: © Copyright 1995 American Mathematical Society