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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
DOI: https://doi.org/10.1090/S0002-9939-99-05314-9
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.


References [Enhancements On Off] (What's this?)

  • [DIX] J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien (Deuxième édition), Gauthier-Villars, Éditeurs, Paris, 1969. MR 50:5482
  • [DO-ST] D.L. Donoho, P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), 906-931. MR 90c:42003
  • [DS] N. Dunford, J.T. Schwarz, Linear Operators (part II), Interscience Publishers, New York, 1963. MR 32:6181
  • [S] K.T. Smith, The uncertainty principle on groups, SIAM J. Appl. Math. 50 (1990), 876-882. MR 91i:94008
  • [STE] G. Stegel, La disuguaglianza di Heisenberg per gruppi liberi, Thesis, Università di Roma ``La Sapienza" (1993).
  • [W] J.A. Wolf, Uncertainty principle for Gelfand pairs, Nova J. Alg. Geom. 1 (1992), 383-396. MR 94k:43006

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Additional Information

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

DOI: https://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

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