设G是局部紧的阿贝尔群，Ω⊂G是Haar测度0＜mG(Ω)＜+∞的一个Borel集。设 g ∈ L2(G)，且 |g|=1/√mG(Ω)1Ω，函数系g(g，∧，(∑))是空间 L2(G)上的一个Gabor系统。以前我们证明了，如果(Q，∧)是一个谱对，(Ω，(∑))是一个tiling对，那么Gabor系统g(g，∧，(∑))是空间L2(G)的一个Gabor正交基。在函数g ∈ L2(G)是非负的条件之下证明上述定理的逆定理。
A Necessary Condition on the Gabor Orthonormal Basis over LCA Groups
Let G be a locally compact Abelien(LCA)group,and let Ω ⊂ G be a Borel set with Haar messure 0＜mG(Ω)＜+∞.Let g ∈ L2(G)with |g|=1/√mG(Ω)1Ω,and let g(g,∧,(∑))be a Gabor system on L2(G).It was proved that if(Ω,∧)is a spectral pair,(Ω,(∑))is a tiling pair,then Q(g,∧,(∑))is a Gabor orthogonal basis for L2(G).In this paper,under the non-negativeness condition of the function g ∈ L2(G),we prove that the inverse of the above theorem is true.
Gabor orthogonal basisnon-negative window functionspectral set conjecture
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