| import Mathlib.Algebra.BigOperators.Group.Finset.Basic | |
| import Mathlib.Logic.Function.Iterate | |
| import Mathlib.Basic.Real.Basic | |
| namespace OAI | |
| noncomputable section | |
| namespace InvariantProjection.Continuity | |
| variable {X : Type*} | |
| def backwardSum (σ : X ≃ X) (v : X → ℝ) (n : ℕ) (x : X) : ℝ := | |
| ∑ r ∈ Finset.range n, v ((σ.symm : X → X)^[r + 1] x) | |
| def relocated (σ : X ≃ X) (h : X → ℝ) (φ : ℕ → X → ℝ) | |
| (H : ℕ) (x : X) : ℝ := | |
| ∑ n ∈ Finset.range (H + 1), | |
| h ((σ.symm : X → X)^[n] x) * φ n ((σ.symm : X → X)^[n] x) | |
| def relocationTransfer (σ : X ≃ X) (h : X → ℝ) (φ : ℕ → X → ℝ) | |
| (H : ℕ) (x : X) : ℝ := | |
| ∑ n ∈ Finset.range (H + 1), backwardSum σ (fun y => h y * φ n y) n x | |
| def backwardAverage (σ : X ≃ X) (v : X → ℝ) (W : ℕ) (x : X) : ℝ := | |
| (W : ℝ)⁻¹ * backwardSum σ v W x | |
| def backgroundTransfer (σ : X ≃ X) (v : X → ℝ) (W : ℕ) (x : X) : ℝ := | |
| (W : ℝ)⁻¹ * ∑ n ∈ Finset.range W, backwardSum σ v (n + 1) x | |
| end InvariantProjection.Continuity | |
| end | |
| end OAI | |