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An Uncertainty principle for convolution operators on discrete groups

Stegel, Giovanni (2000) An Uncertainty principle for convolution operators on discrete groups. Proceedings of the American Mathematical Society, Vol. 128 (6), p. 1807-1812. eISSN 1088-6826. Article.

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DOI: 10.1090/S0002-9939-99-05314-9

Abstract

Consider a discrete group G and a bounded self-adjoint convolution operator T on l2; let σ(T) be the spectrum of T. The spectral theorem gives a unitary isomorphism U between l2(G) and a direct sum ⊕n L2n, ν), where Δn ⊂ σT, and ν is a regular Borel measure supported on σ(T). Through this isomorphism T. corresponds to multiplication by the identity function on each summand. We prove that a nonzero function ƒ ∈ l2(G) and its transform U ƒ cannot be simultaneously concentrated on sets VG, W ⊂ σ(T) such that ν(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.

Item Type:Article
ID Code:5256
Status:Published
Refereed:Yes
Uncontrolled Keywords:Uncertainty principle, discrete groups, convolution operators, Hilbert space
Subjects:Area 01 - Scienze matematiche e informatiche > MAT/05 Analisi matematica
Divisions:001 Università di Sassari > 03 Istituti > Matematica e fisica
Publisher:American Mathematical Society
eISSN:1088-6826
Deposited On:16 Dec 2010 12:38

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