Given a lattice @math in a locally compact abelian group @math and a measurable subset @math with finite and positive measure, then the set of characters associated to the dual lattice form a frame for @math if and only if the distinct translates by @math of @math have almost empty intersections. Some consequences of this results are the well-known Fuglede theorem for lattices, as well as a simple characterization for frames of modulates.
In both papers, @cite_12 and @cite_13 , as well as in the present work, the implication 1) , @math ,4) is done in the same way by observing that @math is an orthogonal basis of exponentials of @math . As for the implication 4) , @math ,1) both papers give a direct proof of the fact that @math have measure zero for all @math and that also the set @math has measure zero. In the present paper we give an argument by contradiction assuming that 1) does not hold and exhibiting a non-zero function in @math which is perpendicular to all @math
@cite_13 @cite_12
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Let Ω denote a connected and open subset of Rn. The existence of n commuting self-adjoint operators H1,…, Hn on L2(Ω) such that each Hj is an extension of −i∂∂xj (acting on Cc∞(Ω)) is shown to be equivalent to the existence of a measure μ on Rn such that f → (the Fourier transform of f) is unitary from L2(Ω) onto Ω. It is shown that the support of μ can be chosen as a subgroup of Rn iff H1,…, Hn can be chosen such that the unitary groups generated by H1,…, Hn act multiplicatively on L2(Ω). This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of Rn by some subgroup, i.e., iff Ω is essentially a fundamental domain. In Rn let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in L2(Ω)). The existence of n commuting self-adjoint operators H1,…, Hn in L2(Ω) such that each Hj is a restriction of −i ββxj (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ⊂Rn such that the restrictions to Ω of the functions exp i ∑ λjxj form a total orthogonal family in L2(Ω). If it is required, in addition, that the unitary groups generated by H1,…, Hn act multiplicatively on L2(Ω), then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rn. The measurable sets Ω ⊂Rn (of finite Lebesgue measure) for which there exists a subgroup Λ ⊂Rn as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rn by some subgroup Γ (essentially the dual of Λ).