By Angus E. Taylor
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Example text
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).
