## Functions on noncompact Lie groups with positive Fourier transforms

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- by Takeshi Kawazoe PDF
- Proc. Amer. Math. Soc.
**123**(1995), 1411-1415 Request permission

## 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

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 1411-1415 - MSC: Primary 42A38; Secondary 22E30, 43A30
- DOI: https://doi.org/10.1090/S0002-9939-1995-1277119-7
- MathSciNet review: 1277119