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An uncertainty principle for convolution operators on discrete groups


Author: Giovanni Stegel
Journal: Proc. Amer. Math. Soc. 128 (2000), 1807-1812
MSC (1991): Primary 43A15, 42A05; Secondary 20F99, 47B37
Published electronically: October 29, 1999
MathSciNet review: 1662222
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Abstract: Consider a discrete group $G$ and a bounded self-adjoint convolution operator $T$ on $l^{2}(G)$; let $\sigma (T)$ be the spectrum of $T$. The spectral theorem gives a unitary isomorphism $U$ between $l^{2}(G)$ and a direct sum $\bigoplus _{n} L^{2}(\Delta _{n},\nu )$, where $\Delta _{n}\subset \sigma (T)$, and $\nu $ is a regular Borel measure supported on $\sigma (T)$. Through this isomorphism $T$ corresponds to multiplication by the identity function on each summand. We prove that a nonzero function $f\in l^{2}(G)$ and its transform $Uf$ cannot be simultaneously concentrated on sets $V\subset G$, $W\subset \sigma (T)$ such that $\nu (W)$ and the cardinality of $V$ are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.


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Additional Information

Giovanni Stegel
Affiliation: Piazza Prati degli Strozzi 35, 00195 Roma, Italy
Email: stegel@marte.mat.uniroma1.it

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05314-9
Keywords: Uncertainty principle, discrete groups, convolution operators, Hilbert space
Received by editor(s): August 1, 1998
Published electronically: October 29, 1999
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society