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7f37411 | 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 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | 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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