Abstract. Consider a discrete group G and a bounded self-adjoint convolution operator T on l 2 (G); let σ(T ) be the spectrum of T . The spectral theorem gives a unitary isomorphism U between l 2 (G) and a direct sum n L 2 (∆n, ν), where ∆n ⊂ σ(T ), 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 f ∈ l 2 (G) and its transform Uf 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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