| import OAI.MeasureTheory.Falconer.Shells.HighShellEndpointTerms |
| import OAI.MeasureTheory.Falconer.Fourier.DistanceKernelSplitting |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ComplexConjugate |
|
|
| theorem integrable_list_sum {α ι : Type*} [MeasurableSpace α] (μ : Measure α) |
| (L : List ι) (f : ι → α → ℂ) (hf : ∀ i ∈ L, Integrable (f i) μ) : |
| Integrable (fun x => (L.map (fun i => f i x)).sum) μ := by |
| induction L with |
| | nil => simp |
| | cons i L ih => |
| simp only [List.map_cons,List.sum_cons] |
| exact (hf i (by simp)).add (ih (fun a ha => hf a (by simp [ha]))) |
|
|
| theorem integral_list_sum {α ι : Type*} [MeasurableSpace α] (μ : Measure α) |
| (L : List ι) (f : ι → α → ℂ) (hf : ∀ i ∈ L, Integrable (f i) μ) : |
| (∫ x, (L.map (fun i => f i x)).sum ∂μ) = (L.map (fun i => ∫ x, f i x ∂μ)).sum := by |
| induction L with |
| | nil => simp |
| | cons i L ih => |
| simp only [List.map_cons,List.sum_cons] |
| rw [integral_add (hf i (by simp)) (integrable_list_sum μ L f (fun a ha => hf a (by simp [ha]))), |
| ih (fun a ha => hf a (by simp [ha]))] |
|
|
| namespace PositiveDistanceData |
|
|
| theorem inverse_angular_kernel_terms (d : PositiveDistanceData) (hQ : 0 < d.Q.side) |
| (j : ℕ) (ψ : UnitCircle → ℂ) (hψ : Measurable ψ) {P : ℝ} (hP : 0 ≤ P) |
| (hp : ∀ dir, ‖ψ dir‖ ≤ P) (r : ℝ) (z : Plane × Plane) : |
| (∫ dir, sourcePlaneWave r dir (z.1-z.2) * |
| (ψ dir * inverseCircleOperator j (d.pairMaskSymbol z) dir) ∂sourceArcLength) = |
| ((inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)).map (fun b => |
| b.1 * ((assignmentScale d.inverseDescriptorList d.N d.sig b.2:ℂ) * |
| sourceDistanceKernel ψ (d.inverseEndpointSymbol b.2 0) |
| (d.inverseEndpointSymbol b.2 1) r z.1 z.2))).sum := by |
| let L := inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList) |
| let f := fun (b : ℂ × (PairMaskIndex → ℕ)) (dir : UnitCircle) => |
| b.1 * ((assignmentScale d.inverseDescriptorList d.N d.sig b.2:ℂ) * |
| (sourcePlaneWave r dir (z.1-z.2) * (ψ dir * d.inverseEndpointSymbol b.2 0 z.1 dir * |
| conj (d.inverseEndpointSymbol b.2 1 z.2 dir)))) |
| have he (dir : UnitCircle) : sourcePlaneWave r dir (z.1-z.2) * |
| (ψ dir * inverseCircleOperator j (d.pairMaskSymbol z) dir) = (L.map (fun b => f b dir)).sum := by |
| rw [d.inverseCircleOperator_pair_terms hQ] |
| rw [← List.sum_map_mul_left, ← List.sum_map_mul_left] |
| apply congrArg List.sum |
| apply List.map_congr_left |
| intro b hb |
| dsimp [f] |
| ring |
| have hi : ∀ b ∈ L, Integrable (f b) sourceArcLength := by |
| intro b hb |
| have hm₁ : Measurable (fun dir => d.inverseEndpointSymbol b.2 0 z.1 dir) := |
| (d.inverseEndpointSymbol_measurable b.2 0).comp |
| (f := fun dir : UnitCircle => (z.1,dir)) (measurable_const.prodMk measurable_id) |
| have hm₂ : Measurable (fun dir => d.inverseEndpointSymbol b.2 1 z.2 dir) := |
| (d.inverseEndpointSymbol_measurable b.2 1).comp |
| (f := fun dir : UnitCircle => (z.2,dir)) (measurable_const.prodMk measurable_id) |
| apply Integrable.const_mul |
| apply Integrable.const_mul |
| refine Integrable.of_bound ?_ |
| (P*(symbolDerivativeConstant (2*j)^(2*j))*(symbolDerivativeConstant (2*j)^(2*j))) ?_ |
| · apply Measurable.aestronglyMeasurable |
| apply Measurable.mul |
| · unfold sourcePlaneWave; fun_prop |
| · exact (hψ.mul hm₁).mul (Complex.continuous_conj.measurable.comp hm₂) |
| · filter_upwards with dir |
| exact distance_integrand_norm_le ψ _ _ hP |
| (pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le _)) _) |
| (pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le _)) _) hp |
| (d.inverseEndpointSymbol_norm hQ j hb 0) (d.inverseEndpointSymbol_norm hQ j hb 1) r dir z.1 z.2 |
| simp_rw [he] |
| rw [integral_list_sum sourceArcLength L f hi] |
| apply congrArg List.sum |
| apply List.map_congr_left |
| intro b hb |
| simp only [f, integral_const_mul, sourceDistanceKernel] |
|
|
| theorem inverseCircleIntegral_kernel_terms (d : PositiveDistanceData) (hQ : 0 < d.Q.side) |
| (n : ℕ) (ψ : UnitCircle → ℂ) (hψ : Measurable ψ) {P : ℝ} (hP : 0 ≤ P) |
| (hp : ∀ dir, ‖ψ dir‖ ≤ P) (r : ℝ) (z : Plane × Plane) : |
| sourceInverseCircleIntegral n r (z.1-z.2) ψ (d.pairMaskSymbol z) = |
| (((r*dist z.1 z.2)^(1/2:ℝ):ℝ):ℂ) * ∑ j ∈ Finset.range (n+1), |
| (((r*dist z.1 z.2)⁻¹:ℝ):ℂ)^j * |
| ((inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)).map (fun b => |
| b.1 * ((assignmentScale d.inverseDescriptorList d.N d.sig b.2:ℂ) * |
| sourceDistanceKernel ψ (d.inverseEndpointSymbol b.2 0) |
| (d.inverseEndpointSymbol b.2 1) r z.1 z.2))).sum := by |
| unfold sourceInverseCircleIntegral |
| simp only [← dist_eq_norm] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro j hj |
| congr 1 |
| exact d.inverse_angular_kernel_terms hQ j ψ hψ hP hp r z |
|
|
| end PositiveDistanceData |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
