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An Uncertainty principle for convolution operators on discrete groups

American Mathematical Society
Publication Date
  • Mat/05 Analisi Matematica
  • Mathematics


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 L2(Δn, 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 V ⊂ G, 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.

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