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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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