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An uncertainty principle for convolution operators on discrete groups

Authors:	Giovanni Stegel

Institution:	Piazza Prati degli Strozzi 35, 00195 Roma, Italy

Abstract:	Consider a discrete group and a bounded self-adjoint convolution operator on $l^{2}(G)$ ; let $\sigma (T)$ be the spectrum of . The spectral theorem gives a unitary isomorphism 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 corresponds to multiplication by the identity function on each summand. We prove that a nonzero function $f\in l^{2}(G)$ and its transform cannot be simultaneously concentrated on sets $V\subset G$ , $W\subset \sigma (T)$ such that $\nu (W)$ and the cardinality of are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

Keywords:	Uncertainty principle discrete groups convolution operators Hilbert space

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