File size: 4,527 Bytes
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 | import OAI.MeasureTheory.Falconer.Shells.HighShellFiniteSeparation
import OAI.MeasureTheory.Falconer.Shells.ShellMultipliers
import OAI.MeasureTheory.Falconer.Estimates.SpatialPhasedDistanceForm
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
namespace PositiveDistanceData
noncomputable def inversePhasedForm (d : PositiveDistanceData)
(k : PairMaskIndex → ℕ) (ξ : Plane) : ℝ → ℂ :=
sourceParentDistanceF (d.Q.descendant d.X) (d.Q.descendant d.Y)
d.ρ d.ρ d.levelMass d.levelMass
(spatialPhasedSymbol ((2:ℝ)^(-(d.a:ℝ))) ξ (d.inverseEndpointSymbol k 0))
(spatialPhasedSymbol ((2:ℝ)^(-(d.a:ℝ))) ξ (d.inverseEndpointSymbol k 1))
noncomputable def inverseSpatialMixture (d : PositiveDistanceData) (j : ℕ)
(k : PairMaskIndex → ℕ) (r : ℝ) : ℂ :=
∫ ξ, spatialFourierCoeff 96 2004 (by norm_num) (by norm_num) j ξ * d.inversePhasedForm k ξ r
noncomputable def inversePhasedBound (d : PositiveDistanceData) (j : ℕ) : ℝ :=
(1*(symbolDerivativeConstant (2*j)^(2*j))*(symbolDerivativeConstant (2*j)^(2*j))*
sourceArcLength.real univ) * ((d.ρ (d.Q.descendant d.X).carrier).toReal/d.levelMass)*
((d.ρ (d.Q.descendant d.Y).carrier).toReal/d.levelMass)
theorem inversePhasedBound_nonneg (d : PositiveDistanceData) (j : ℕ) :
0 ≤ d.inversePhasedBound j := by
have hC := zero_le_one.trans (symbolDerivativeConstant_one_le (2*j))
unfold inversePhasedBound levelMass
positivity
theorem inversePhasedForm_data (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (j : ℕ) {b : ℂ × (PairMaskIndex → ℕ)}
(hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)) :
Measurable (fun p : Plane × ℝ => d.inversePhasedForm b.2 p.1 p.2) ∧
∀ p : Plane × ℝ, ‖d.inversePhasedForm b.2 p.1 p.2‖ ≤ d.inversePhasedBound j := by
have hC : 0 ≤ symbolDerivativeConstant (2*j)^(2*j) :=
pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le _)) _
constructor
· exact spatialPhasedDistanceF_joint_measurable _ d.ρ d.ρ _ _ d.levelMass d.levelMass
_ _ _ (measurable_sourceConeCircle _) (d.inverseEndpointSymbol_measurable b.2 0)
(d.inverseEndpointSymbol_measurable b.2 1) (by norm_num) hC hC
(sourceConeCircle_norm_le _) (d.inverseEndpointSymbol_norm hQ j hb 0)
(d.inverseEndpointSymbol_norm hQ j hb 1)
· intro p
apply sourceDistanceF_norm_le d.ρ d.ρ _ _
(show 0 < d.levelMass by unfold levelMass; positivity)
(show 0 < d.levelMass by unfold levelMass; positivity)
_ _ _ (by norm_num) hC hC (sourceConeCircle_norm_le _)
· intro x hx ω
rw [spatialPhasedSymbol_norm]
exact d.inverseEndpointSymbol_norm hQ j hb 0 x ω
· intro y hy ω
rw [spatialPhasedSymbol_norm]
exact d.inverseEndpointSymbol_norm hQ j hb 1 y ω
theorem inverseSpatialMixture_data (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (j : ℕ) {b : ℂ × (PairMaskIndex → ℕ)}
(hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList))
(μ : Measure ℝ) [IsFiniteMeasure μ] :
Measurable (d.inverseSpatialMixture j b.2) ∧
(∀ r, ‖d.inverseSpatialMixture j b.2 r‖ ≤
spatialFourierCost 96 2004 (by norm_num) (by norm_num) j*d.inversePhasedBound j) ∧
Integrable (fun r => ‖d.inverseSpatialMixture j b.2 r‖^2) μ := by
have hd := d.inversePhasedForm_data hQ j hb
exact continuous_shell_mixture_bound μ _
(spatialFourierCoeff 96 2004 (by norm_num) (by norm_num) j).continuous.measurable
(spatialFourierCoeff_integrable _ _ _ _ _) _ hd.1 (d.inversePhasedBound_nonneg j) hd.2
theorem inverseSpatialMixture_shell (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (j : ℕ) {b : ℂ × (PairMaskIndex → ℕ)}
(hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList))
(μ : Measure ℝ) [IsFiniteMeasure μ] {B : ℝ}
(hB : ∀ ξ, (∫ r, ‖d.inversePhasedForm b.2 ξ r‖^2 ∂μ) ≤ B) :
(∫ r, ‖d.inverseSpatialMixture j b.2 r‖^2 ∂μ) ≤
(spatialFourierCost 96 2004 (by norm_num) (by norm_num) j)^2*B := by
have hd := d.inversePhasedForm_data hQ j hb
exact continuous_coefficient_shell_square μ _
(spatialFourierCoeff 96 2004 (by norm_num) (by norm_num) j).continuous.measurable
(spatialFourierCoeff_integrable _ _ _ _ _) _ hd.1 (d.inversePhasedBound_nonneg j) hd.2 hB
end PositiveDistanceData
end PlanarFalconer
end OAI
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