File size: 3,452 Bytes
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 | 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
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