By Mads Sielemann Jakobsen, Jakob Lemvig
We examine Gabor frames on in the neighborhood compact abelian teams with time–frequency shifts alongside non-separable, closed subgroups of the part area. Density theorems in Gabor research nation valuable stipulations for a Gabor method to be a body or a Riesz foundation, formulated in basic terms by way of the index subgroup. within the classical effects the subgroup is thought to be discrete. We end up density theorems for common closed subgroups of the section area, the place the required stipulations are given when it comes to the “size” of the subgroup. From those density effects we can expand the classical Wexler–Raz biorthogonal relatives and the duality precept in Gabor research to Gabor structures with time–frequency shifts alongside non-separable, closed subgroups of the part house. Even within the euclidean atmosphere, our effects are new.
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Additional info for Density and duality theorems for regular Gabor frames
1) F Vg f · Vg Eω Tx f (β, αx) d(β, α) d(x, ω) = G×G G×G (c) f, Eβω Tα−1 f Eβ Tx−1 α−1 g, g d(β, α) d(x, ω) = G×G G×G α → α−1 , β → βω −1 = f, Eβ Tα f Eβω−1 Tαx−1 g, g d(β, α) d(x, ω) G×G G×G |Vf f (α, β)| = G×G = f S0 ,f |Vg g(αx−1 , βω −1 )| d(x, ω) d(α, β) G×G g S0 ,g . 4. Let fi ∈ L2 (G) and gi ∈ S0 (G), i = 1, 2. Then the mapping ϕ : G × G → C, (x, ω) → Vg1 f1 · Vg2 f2 (x, ω) belongs to S0 (G × G). Proof. It is clear that ϕ ∈ L1 (G × G). Now, let g0 ∈ S0 (G) and deﬁne ϕ0 := Vg0 g0 . 3 the function ϕ0 ∈ S0 (G × G), and thus ϕ20 ∈ S0 (G × G).
Lemvig, Reproducing formulas for generalized translation invariant systems on locally compact abelian groups, Trans. Amer. Math. Soc. 4948, in press. S. Jakobsen, J. Lemvig, Co-compact Gabor systems on locally compact abelian groups, J. Fourier Anal. 1007/s00041-015-9407-0, arXiv:1411:4948, in press. M. Janssen, Duality and biorthogonality for Weyl–Heisenberg frames, J. Fourier Anal. Appl. 1 (4) (1995) 403–436. A. Johnson, Atomic and nonatomic measures, Proc. Amer. Math. Soc. 25 (1970) 650–655.
Proof. Let f, g ∈ S0 (G). By the argument preceding the proposition, we have that Vg f ∈ L1 (G × G). 1) F Vg f · Vg Eω Tx f (β, αx) d(β, α) d(x, ω) = G×G G×G (c) f, Eβω Tα−1 f Eβ Tx−1 α−1 g, g d(β, α) d(x, ω) = G×G G×G α → α−1 , β → βω −1 = f, Eβ Tα f Eβω−1 Tαx−1 g, g d(β, α) d(x, ω) G×G G×G |Vf f (α, β)| = G×G = f S0 ,f |Vg g(αx−1 , βω −1 )| d(x, ω) d(α, β) G×G g S0 ,g . 4. Let fi ∈ L2 (G) and gi ∈ S0 (G), i = 1, 2. Then the mapping ϕ : G × G → C, (x, ω) → Vg1 f1 · Vg2 f2 (x, ω) belongs to S0 (G × G).