| import OAI.MeasureTheory.Falconer.Measures.NormalizedLeibnizAssignment |
| import OAI.MeasureTheory.Falconer.Estimates.InverseLeibniz |
| import OAI.MeasureTheory.Falconer.Fourier.MinimalPairSymbol |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ContDiff ComplexConjugate |
|
|
| abbrev PairMaskIndex := PolicyDescriptor × Fin 2 |
|
|
| noncomputable def pairMaskIndices (S : Finset PolicyDescriptor) : Finset PairMaskIndex := |
| S.product Finset.univ |
|
|
| noncomputable def assignmentScale (S : Finset PolicyDescriptor) (N : ℕ) (sig : ℝ) |
| (k : PairMaskIndex → ℕ) : ℝ := |
| ∏ a ∈ pairMaskIndices S, (sourceAngularRadius N sig (descriptorLength a.1))⁻¹^(k a) |
|
|
| theorem paired_inheritedMaskProduct (S : Finset PolicyDescriptor) (k : PairMaskIndex → ℕ) |
| (Q : DyadicSquare) (ρ : Measure Plane) (m : ℝ → ℝ) (N : ℕ) (lam sig : ℝ) (J i : ℕ) |
| (x : Fin 2 → Plane) (θ : ℝ) : |
| (inheritedMaskProduct (pairMaskIndices S) Prod.fst k Q ρ m N lam sig J i (fun a => x a.2) θ : ℂ) = |
| inheritedCircleSymbol S id (fun b => k (b,0)) Q ρ m N lam sig J i (fun _ => 1) (x 0) (sourceUnitDirection θ) * |
| conj (inheritedCircleSymbol S id (fun b => k (b,1)) Q ρ m N lam sig J i (fun _ => 1) (x 1) (sourceUnitDirection θ)) := by |
| rw [inheritedCircleSymbol_comp,inheritedCircleSymbol_comp] |
| simp only [one_mul,Complex.conj_ofReal,← Complex.ofReal_mul] |
| congr 1 |
| simp [inheritedMaskProduct,pairMaskIndices,Finset.prod_product,Fin.prod_univ_two, |
| Finset.prod_mul_distrib] |
|
|
| theorem paired_assignment_normalized (S : Finset PolicyDescriptor) (k : PairMaskIndex → ℕ) |
| (Q : DyadicSquare) (hQ : 0 < Q.side) (ρ : Measure Plane) (m : ℝ → ℝ) |
| (N : ℕ) (lam sig : ℝ) (J i : ℕ) (x : Fin 2 → Plane) (θ : ℝ) : |
| assignedDerivProduct (pairMaskIndices S) |
| (fun a t => (descriptorAngleMask Q ρ m N lam sig J i a.1 (x a.2) t : ℂ)) k θ = |
| (assignmentScale S N sig k : ℂ) * |
| (inheritedCircleSymbol S id (fun b => k (b,0)) Q ρ m N lam sig J i (fun _ => 1) (x 0) (sourceUnitDirection θ) * |
| conj (inheritedCircleSymbol S id (fun b => k (b,1)) Q ρ m N lam sig J i (fun _ => 1) (x 1) (sourceUnitDirection θ))) := by |
| rw [actual_assignment_normalized _ _ _ _ hQ,paired_inheritedMaskProduct] |
| rfl |
|
|
| namespace PositiveDistanceData |
|
|
| theorem pairMaskSymbol_product (d : PositiveDistanceData) (z : Plane × Plane) (θ : ℝ) : |
| let S := policyDescriptorList (fun n => d.mass n-(n:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t |
| d.pairMaskSymbol z (sourceUnitDirection θ) = |
| ∏ a ∈ pairMaskIndices S, |
| (descriptorAngleMask d.Q d.ρ d.mass d.N d.lam d.sig d.J d.i a.1 (![z.1,z.2] a.2) θ : ℂ) := by |
| simp [pairMaskSymbol,pairMaskIndices,Finset.prod_product,Fin.prod_univ_two, |
| Finset.prod_mul_distrib,descriptorAngleMask,descriptorListMask] |
|
|
| theorem pairMaskSymbol_inverse_leibniz (d : PositiveDistanceData) (hQ : 0 < d.Q.side) |
| (z : Plane × Plane) (j : ℕ) (θ : ℝ) : |
| let S := policyDescriptorList (fun n => d.mass n-(n:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t |
| inverseStationaryOperator j (fun t => d.pairMaskSymbol z (sourceUnitDirection t)) θ = |
| ((inverseLeibnizTerms j (pairMaskIndices S)).map (fun a => |
| a.1 * ((assignmentScale S d.N d.sig a.2:ℂ) * |
| (inheritedCircleSymbol S id (fun b => a.2 (b,0)) d.Q d.ρ d.mass d.N d.lam d.sig d.J d.i (fun _ => 1) z.1 (sourceUnitDirection θ) * |
| conj (inheritedCircleSymbol S id (fun b => a.2 (b,1)) d.Q d.ρ d.mass d.N d.lam d.sig d.J d.i (fun _ => 1) z.2 (sourceUnitDirection θ)))))).sum := by |
| dsimp only |
| let S := policyDescriptorList (fun n => d.mass n-(n:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t |
| let f : PairMaskIndex → ℝ → ℂ := fun a t => |
| (descriptorAngleMask d.Q d.ρ d.mass d.N d.lam d.sig d.J d.i a.1 (![z.1,z.2] a.2) t : ℂ) |
| have hf : ∀ a ∈ pairMaskIndices S, ContDiff ℝ ∞ (f a) := by |
| intro a ha |
| exact Complex.ofRealCLM.contDiff.comp |
| (descriptorAngleMask_contDiff d.Q hQ d.ρ d.mass d.N d.lam d.sig d.J d.i a.1 (![z.1,z.2] a.2)) |
| simp_rw [pairMaskSymbol_product] |
| change inverseStationaryOperator j (fun t => ∏ a ∈ pairMaskIndices S, f a t) θ = _ |
| rw [inverseLeibnizTerms_expansion j (pairMaskIndices S) f hf] |
| apply congrArg List.sum |
| apply List.map_congr_left |
| intro a ha |
| change a.1 * assignedDerivProduct (pairMaskIndices S) |
| (fun b t => (descriptorAngleMask d.Q d.ρ d.mass d.N d.lam d.sig d.J d.i b.1 (![z.1,z.2] b.2) t : ℂ)) a.2 θ = _ |
| rw [paired_assignment_normalized _ _ _ hQ] |
| rfl |
|
|
| end PositiveDistanceData |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
