A qualitative uncertainty principle for unimodular groups of type 𝐼

Hogan, Jeffrey A.

doi:10.1090/S0002-9947-1993-1102222-4

A qualitative uncertainty principle for unimodular groups of type $\textrm {I}$
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by Jeffrey A. Hogan

Trans. Amer. Math. Soc. 340 (1993), 587-594

DOI: https://doi.org/10.1090/S0002-9947-1993-1102222-4

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Abstract:

It has long been known that if $f \in {L^2}({{\mathbf {R}}^n})$ and the supports of and its Fourier transform $\hat f$ are bounded then $f = 0$ almost everywhere. More recently it has been shown that the same conclusion can be reached under the weaker condition that the supports of $f$ and $\hat f$ have finite measure. These results may be thought of as qualitative uncertainty principles since they limit the "concentration" of the Fourier transform pair $(f,\hat f)$. Little is known, however, of analogous results for functions on locally compact groups. A qualitative uncertainty principle is proved here for unimodular groups of type I.

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Harmonic analysis on semisimple Lie groups