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A COMPACT QUALITATIVE UNCERTAINTY PRINCIPLE FOR SOME NONUNIMODULAR GROUPS
- Source :
- Bulletin of the Australian Mathematical Society. 99:114-120
- Publication Year :
- 2018
- Publisher :
- Cambridge University Press (CUP), 2018.
-
Abstract
- Let $G$ be a separable locally compact group with type $I$ left regular representation, $\widehat{G}$ its dual, $A(G)$ its Fourier algebra and $f\in A(G)$ with compact support. If $G=\mathbb{R}$ and the Fourier transform of $f$ is compactly supported, then, by a classical Paley–Wiener theorem, $f=0$. There are extensions of this theorem for abelian and some unimodular groups. In this paper, we prove that if $G$ has no (nonempty) open compact subsets, $\hat{f}$, the regularised Fourier cotransform of $f$, is compactly supported and $\text{Im}\,\hat{f}$ is finite dimensional, then $f=0$. In connection with this result, we characterise locally compact abelian groups whose identity components are noncompact.
- Subjects :
- Pure mathematics
Uncertainty principle
Fourier algebra
General Mathematics
010102 general mathematics
0102 computer and information sciences
01 natural sciences
Connection (mathematics)
Harmonic analysis
symbols.namesake
Identity (mathematics)
Fourier transform
010201 computation theory & mathematics
symbols
Locally compact space
0101 mathematics
Abelian group
Mathematics
Subjects
Details
- ISSN :
- 17551633 and 00049727
- Volume :
- 99
- Database :
- OpenAIRE
- Journal :
- Bulletin of the Australian Mathematical Society
- Accession number :
- edsair.doi...........ee3d34f5e0b2c901e9fcd6c96f83479a