Functions on noncompact Lie groups with positive Fourier transforms

Kawazoe, Takeshi

doi:10.1090/S0002-9939-1995-1277119-7

Functions on noncompact Lie groups with positive Fourier transforms
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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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