Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
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?)

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

PII: S 0002-9939(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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia