Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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] Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (algèbres de von Neumann), Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition, revue et augmentée; Cahiers Scientifiques, Fasc. XXV. MR 0352996
  • [DO-ST] David L. Donoho and Philip B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), no. 3, 906–931. MR 997928, 10.1137/0149053
  • [DS] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745
  • [S] Kennan T. Smith, The uncertainty principle on groups, SIAM J. Appl. Math. 50 (1990), no. 3, 876–882. MR 1050918, 10.1137/0150051
  • [STE] G. Stegel, La disuguaglianza di Heisenberg per gruppi liberi, Thesis, Università di Roma ``La Sapienza" (1993).
  • [W] Joseph A. Wolf, The uncertainty principle for Gel′fand pairs, Nova J. Algebra Geom. 1 (1992), no. 4, 383–396. MR 1218362

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 43A15, 42A05, 20F99, 47B37

Retrieve articles in all journals with MSC (1991): 43A15, 42A05, 20F99, 47B37

Additional Information

Giovanni Stegel
Affiliation: Piazza Prati degli Strozzi 35, 00195 Roma, Italy

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