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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Analysis.IrrationalRotation.Model.ConditionalConvolution
import OAI.Analysis.IrrationalRotation.Analysis.PointwiseConvolution
namespace OAI
open scoped ComplexConjugate ENNReal lp
namespace InvariantProjection.ConditionalConvolution
theorem tsum_omittedEvent {Ω : Type*} (H : ℤ → Set Ω) (K n : ℕ)
(ω : Ω) (w : ℤ → ℝ) :
(∑' v : ↑(((window K n) : Set ℤ)ᶜ),
(H (selector K n v)).indicator (fun _ => (1 : ℝ)) ω * w v) =
∑' v : ℤ, (omittedEvent H K n v).indicator (fun _ => (1 : ℝ)) ω * w v := by
classical
rw [tsum_subtype (((window K n) : Set ℤ)ᶜ)
(fun v : ℤ => (H (selector K n v)).indicator (fun _ => (1 : ℝ)) ω * w v)]
apply tsum_congr
intro v
by_cases hv : v ∈ window K n <;> simp [omittedEvent, hv]
theorem pointwise_bound {Ω : Type*} (K n : ℕ) (Fx Fy : ℤ → ℝ)
(ξ γ : ℓ²(ℤ, ℂ)) (hξ : ‖ξ‖ ≤ 1) (hγ : ‖γ‖ ≤ 1)
(η : ℝ) (hη : 0 ≤ η) (H₁ H₂ : ℤ → Set Ω) (ω : Ω)
(hratio : ∀ v : ℤ, 0 < Fy ((n : ℤ) + v) / Fx v ∧
Fy ((n : ℤ) + v) / Fx v ≤ min (Fy v / Fx v)
(Fy ((n : ℤ) + v) / Fx ((n : ℤ) + v)) ∧
min (Fy v / Fx v) (Fy ((n : ℤ) + v) / Fx ((n : ℤ) + v)) ≤ 1)
(hzero : (∑' v : ℤ, (ξ v / (Fx v : ℂ)) *
((Fy ((n : ℤ) + v) : ℂ) * conj (γ ((n : ℤ) + v)))) = 0)
(htail : ∀ k : ℤ, (K : ℤ) < |k| → Fy k / Fx k ≤ η +
(H₁ k).indicator (fun _ => (1 : ℝ)) ω +
(H₂ k).indicator (fun _ => (1 : ℝ)) ω) :
‖truncated K n (fun v => ξ v / (Fx v : ℂ))
(fun w => (Fy (-w) : ℂ) * conj (γ (-w)))‖ ≤ η +
(∑' v : ℤ, (omittedEvent H₁ K n v).indicator (fun _ => (1 : ℝ)) ω *
(‖ξ v‖ * ‖γ ((n : ℤ) + v)‖)) +
(∑' v : ℤ, (omittedEvent H₂ K n v).indicator (fun _ => (1 : ℝ)) ω *
(‖ξ v‖ * ‖γ ((n : ℤ) + v)‖)) := by
classical
have hmask (H : ℤ → Set Ω) (k : ℤ) :
({k | ω ∈ H k} : Set ℤ).indicator (fun _ => (1 : ℝ)) k =
(H k).indicator (fun _ => (1 : ℝ)) ω := by
simp only [Set.indicator_apply, Set.mem_ofPred_eq]
have hdom := InvariantProjection.l2_masked_convolution_tail_selector
(window K n) ξ γ (Equiv.addLeft (n : ℤ)) hξ hγ
(fun v => Fy ((n : ℤ) + v) / Fx v)
(fun v => ⟨(hratio v).1.le, (hratio v).2.1.trans (hratio v).2.2⟩)
η hη {k | ω ∈ H₁ k} {k | ω ∈ H₂ k} (selector K n)
(by
change (∑' v : ℤ, ((Fy ((n : ℤ) + v) / Fx v : ℝ) : ℂ) * ξ v *
conj (γ ((n : ℤ) + v))) = 0
calc
_ = ∑' v : ℤ, (ξ v / (Fx v : ℂ)) *
((Fy ((n : ℤ) + v) : ℂ) * conj (γ ((n : ℤ) + v))) := by
apply tsum_congr
intro v
exact (InvariantProjection.convolution_coefficient_factorization Fx Fy
(fun v => ξ v) (fun v => γ v) (n : ℤ) v).symm
_ = 0 := hzero)
(by
intro v hv
have hvw : ¬ (|v| ≤ (K : ℤ) ∧ |(n : ℤ) + v| ≤ (K : ℤ)) := by
intro hw
apply hv
simp only [window, Finset.mem_filter, Finset.mem_Icc]
exact ⟨abs_le.mp hw.1, hw.2⟩
have hk := InvariantProjection.omitted_index_selector (K : ℤ) (n : ℤ) v hvw
have hb := (InvariantProjection.omitted_index_ratio_bound (K : ℤ) (n : ℤ) v
(fun k => Fy k / Fx k) _ (hratio v).2.1).trans (htail _ hk)
simpa only [selector, hmask] using hb)
simp only [hmask] at hdom
rw [tsum_omittedEvent H₁ K n ω
(fun v => ‖ξ v‖ * ‖γ ((Equiv.addLeft (n : ℤ)) v)‖),
tsum_omittedEvent H₂ K n ω
(fun v => ‖ξ v‖ * ‖γ ((Equiv.addLeft (n : ℤ)) v)‖)] at hdom
convert hdom using 1
congr 1
apply Finset.sum_congr rfl
intro v hv
change (ξ v / (Fx v : ℂ)) *
((Fy (-(-(n : ℤ) - v)) : ℂ) * conj (γ (-(-(n : ℤ) - v)))) = _
have hsign : -(-(n : ℤ) - v) = (n : ℤ) + v := by omega
rw [hsign]
exact InvariantProjection.convolution_coefficient_factorization Fx Fy (fun v => ξ v) (fun v => γ v) (n : ℤ) v
all_goals rfl
end InvariantProjection.ConditionalConvolution
end OAI
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