By Mads Sielemann Jakobsen, Jakob Lemvig
We examine Gabor frames on in the community compact abelian teams with time–frequency shifts alongside non-separable, closed subgroups of the section house. Density theorems in Gabor research country helpful stipulations for a Gabor process to be a body or a Riesz foundation, formulated in simple terms when it comes to the index subgroup. within the classical effects the subgroup is believed to be discrete. We end up density theorems for normal closed subgroups of the section house, the place the mandatory stipulations are given by way of the “size” of the subgroup. From those density effects we will expand the classical Wexler–Raz biorthogonal kin and the duality precept in Gabor research to Gabor structures with time–frequency shifts alongside non-separable, closed subgroups of the section house. Even within the euclidean environment, our effects are new.
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Extra info for Density and duality theorems for regular Gabor frames
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2. Let g, gi , f, fi ∈ L2 (G), i = 1, 2 and x, α ∈ G and ω, β ∈ G. Then the short-time Fourier transform Vg : L2 (G) → L2 (G × G), Vg f (x, ω) = f, Eω Tx g satisﬁes the following relations: (a) Vg Eβ Tα f = β(α) E(eG ,α−1 ) T(α,β) Vg f , where eG denotes the identity element in G, (b) VEβ Tα g Eβ Tα f = β(x)ω(α) Vg f , (c) F(Vg1 f1 ·Vg2 f2 )(β, α) = f1 , Eβ Tα−1 f2 Eβ Tα−1 g2 , g1 , where F is the Fourier transform on G × G. Proof. Assertion (a) follows from: (Vg Eβ Tα f )(x, ω) = Eβ Tα f, Eω Tx g = f, Tα−1 Eωβ −1 Tx g = ω(α)β(α) f, Eωβ −1 Txα−1 g = ω(α)β(α)Vg f (xα−1 , ωβ −1 ) = β(α) E(eG ,α−1 ) T(α,β) Vg f (x, ω).
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