| import Mathlib.Data.Int.Interval |
| import Mathlib.Basic.Real.Basic |
| import Mathlib.Analysis.Complex.Basic |
| import Mathlib.Analysis.InnerProductSpace.l2Space |
| import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic |
|
|
| namespace OAI |
|
|
| open MeasureTheory |
| open scoped ENNReal lp ComplexConjugate |
|
|
| noncomputable section |
|
|
| namespace InvariantProjection.ConditionalConvolution |
|
|
| def window (K : ℕ) (n : ℕ) : Finset ℤ := |
| (Finset.Icc (-(K : ℤ)) (K : ℤ)).filter fun v => |(n : ℤ) + v| ≤ (K : ℤ) |
|
|
| def selector (K : ℕ) (n : ℕ) (v : ℤ) : ℤ := |
| if (K : ℤ) < |v| then v else (n : ℤ) + v |
|
|
| def omittedEvent {Ω : Type*} (H : ℤ → Set Ω) (K : ℕ) (n : ℕ) (v : ℤ) : Set Ω := |
| if v ∈ window K n then ∅ else H (selector K n v) |
|
|
| def truncated (K : ℕ) (n : ℕ) (A B : ℤ → ℂ) : ℂ := |
| ∑ v ∈ window K n, A v * B (-(n : ℤ) - v) |
|
|
| def coeffA {Ω : Type*} (ξ : Ω → ℓ²(ℤ, ℂ)) (Fx : Ω → ℤ → ℝ) |
| (ω : Ω) (v : ℤ) : ℂ := ξ ω v / (Fx ω v : ℂ) |
|
|
| def coeffB {Ω : Type*} (γ : Ω → ℓ²(ℤ, ℂ)) (Fy : Ω → ℤ → ℝ) |
| (ω : Ω) (w : ℤ) : ℂ := (Fy ω (-w) : ℂ) * conj (γ ω (-w)) |
|
|
| structure Data (Ω : Type*) [mΩ : MeasurableSpace Ω] (μ : Measure Ω) |
| (K : ℕ) (D a c η δ : ℝ) where |
| mX : MeasurableSpace Ω |
| mY : MeasurableSpace Ω |
| mX_le : mX ≤ mΩ |
| mY_le : mY ≤ mΩ |
| ξ : Ω → ℓ²(ℤ, ℂ) |
| γ : Ω → ℓ²(ℤ, ℂ) |
| ξ_measurable : StronglyMeasurable[mX] ξ |
| γ_measurable : StronglyMeasurable[mY] γ |
| ξ_unit : ∀ᵐ ω ∂μ, ‖ξ ω‖ ≤ 1 |
| γ_unit : ∀ᵐ ω ∂μ, ‖γ ω‖ ≤ 1 |
| Fx : Ω → ℤ → ℝ |
| Fy : Ω → ℤ → ℝ |
| Fx_measurable : ∀ k, Measurable[mΩ] (fun ω => Fx ω k) |
| Fy_measurable : ∀ k, Measurable[mΩ] (fun ω => Fy ω k) |
| Fx_pos : ∀ᵐ ω ∂μ, ∀ k, 0 < Fx ω k |
| Fy_pos : ∀ᵐ ω ∂μ, ∀ k, 0 < Fy ω k |
| Fx_zero : ∀ᵐ ω ∂μ, Fx ω 0 = 1 |
| Fy_zero : ∀ᵐ ω ∂μ, Fy ω 0 = 1 |
| zero_convolution : ∀ n : ℕ, n ≤ 2 * K → ∀ᵐ ω ∂μ, |
| (∑' v : ℤ, coeffA ξ Fx ω v * coeffB γ Fy ω (-(n : ℤ) - v)) = 0 |
| ratio_bounds : ∀ n : ℕ, n ≤ 2 * K → ∀ᵐ ω ∂μ, ∀ v : ℤ, |
| 0 < Fy ω ((n : ℤ) + v) / Fx ω v ∧ |
| Fy ω ((n : ℤ) + v) / Fx ω v ≤ |
| min (Fy ω v / Fx ω v) (Fy ω ((n : ℤ) + v) / Fx ω ((n : ℤ) + v)) ∧ |
| min (Fy ω v / Fx ω v) (Fy ω ((n : ℤ) + v) / Fx ω ((n : ℤ) + v)) ≤ 1 |
| E : Set Ω |
| E_measurable : MeasurableSet[mΩ] E |
| E_mass : c ≤ μ.real E |
| coefficient_bounds : ∀ᵐ ω ∂μ.restrict E, |
| (∀ v : ℤ, |v| ≤ (K : ℤ) → ‖coeffA ξ Fx ω v‖ ≤ D) ∧ |
| (∀ v : ℤ, |v| ≤ (K : ℤ) → ‖coeffB γ Fy ω v‖ ≤ D) ∧ |
| a ≤ ‖coeffA ξ Fx ω 0‖ ∧ a ≤ ‖coeffB γ Fy ω 0‖ |
| H₁ : ℤ → Set Ω |
| H₂ : ℤ → Set Ω |
| H₁_measurable : ∀ k, MeasurableSet[mΩ] (H₁ k) |
| H₂_measurable : ∀ k, MeasurableSet[mΩ] (H₂ k) |
| attenuation : ∀ᵐ ω ∂μ.restrict E, ∀ k : ℤ, (K : ℤ) < |k| → |
| Fy ω k / Fx ω k ≤ η + (H₁ k).indicator (fun _ => (1 : ℝ)) ω + |
| (H₂ k).indicator (fun _ => (1 : ℝ)) ω |
| conditional₁ : ∀ k : ℤ, (K : ℤ) < |k| → |
| μ[(H₁ k).indicator (fun _ => (1 : ℝ)) | mX] ≤ᵐ[μ] fun _ => δ |
| conditional₂ : ∀ k : ℤ, (K : ℤ) < |k| → |
| μ[(H₂ k).indicator (fun _ => (1 : ℝ)) | mY] ≤ᵐ[μ] fun _ => δ |
|
|
| end InvariantProjection.ConditionalConvolution |
|
|
| end |
|
|
| end OAI |
|
|
