File size: 4,739 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 | import OAI.MeasureTheory.Falconer.Shells.HighShellFourierError
import OAI.MeasureTheory.Falconer.Geometry.SeparatedSpatialDistanceForm
import OAI.MeasureTheory.Falconer.Graphs.ParentRetainedEnergy
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory Filter
namespace PositiveDistanceData
noncomputable def inverseDistanceEndpointKernel (d : PositiveDistanceData) (j : ℕ)
(k : PairMaskIndex → ℕ) (r : ℝ) (z : Plane × Plane) : ℂ :=
((((dist z.1 z.2/(2:ℝ)^(-(d.a:ℝ)))⁻¹)^j:ℝ):ℂ) *
sourceDistanceKernel
(sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y)))
(d.inverseEndpointSymbol k 0) (d.inverseEndpointSymbol k 1) r z.1 z.2
theorem inverseDistanceEndpointKernel_measurable (d : PositiveDistanceData) (j : ℕ)
(k : PairMaskIndex → ℕ) (r : ℝ) : Measurable (d.inverseDistanceEndpointKernel j k r) := by
unfold inverseDistanceEndpointKernel
apply Measurable.mul
· fun_prop
· exact (measurable_sourceDistanceKernel _ _ _ (measurable_sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y)))
(d.inverseEndpointSymbol_measurable k 0) (d.inverseEndpointSymbol_measurable k 1)).comp
(f := fun z : Plane × Plane => (r,z)) (measurable_const.prodMk measurable_id)
theorem inverseDistanceEndpointKernel_integrable (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (hs : d.StronglySeparated) (j : ℕ)
{b : ℂ × (PairMaskIndex → ℕ)} (hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList))
(r : ℝ) : Integrable (d.inverseDistanceEndpointKernel j b.2 r) d.cellPair := by
let C := symbolDerivativeConstant (2*j)^(2*j)
have hC : 0 ≤ C := pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le _)) _
refine Integrable.of_bound (d.inverseDistanceEndpointKernel_measurable j b.2 r).aestronglyMeasurable
(1*C*C*sourceArcLength.real univ) ?_
filter_upwards [d.ae_cellPair_mem] with z hz
have hd := (d.separated_distance_ratio hQ hs (z := z) hz.1 hz.2).1
have hdist : 1 ≤ dist z.1 z.2/(2:ℝ)^(-(d.a:ℝ)) := by linarith
have hinv : (dist z.1 z.2/(2:ℝ)^(-(d.a:ℝ)))⁻¹ ≤ 1 := inv_le_one_of_one_le₀ hdist
have hn : ‖((((dist z.1 z.2/(2:ℝ)^(-(d.a:ℝ)))⁻¹)^j:ℝ):ℂ)‖ ≤ 1 := by
rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (by positivity)]
exact pow_le_one₀ (by positivity) hinv
have hk := sourceDistanceKernel_norm_le _ _ _ (by norm_num : (0:ℝ) ≤ 1) hC hC
(sourceConeCircle_norm_le (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y))) (d.inverseEndpointSymbol_norm hQ j hb 0)
(d.inverseEndpointSymbol_norm hQ j hb 1) r z.1 z.2
change ‖_ * _‖ ≤ _
rw [norm_mul]
exact (mul_le_mul_of_nonneg_right hn (norm_nonneg _)).trans (by simpa only [one_mul] using hk)
theorem inverseDistanceEndpointKernel_form (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (hs : d.StronglySeparated) (j : ℕ)
{b : ℂ × (PairMaskIndex → ℕ)} (hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList))
(r : ℝ) :
(d.levelMass^(-2:ℤ):ℝ) • (∫ z, d.inverseDistanceEndpointKernel j b.2 r z ∂d.cellPair) =
∫ ξ, spatialFourierCoeff 96 2004 (by norm_num) (by norm_num) j ξ *
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 b.2 0))
(spatialPhasedSymbol ((2:ℝ)^(-(d.a:ℝ))) ξ (d.inverseEndpointSymbol b.2 1)) r := by
have hC : 0 ≤ symbolDerivativeConstant (2*j)^(2*j) :=
pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le _)) _
have he := separatedSpatialDistanceForm 96 2004 (by norm_num) (by norm_num) j
(s := (2:ℝ)^(-(d.a:ℝ))) (by positivity) d.ρ d.ρ
(d.Q.descendant d.X).carrier (d.Q.descendant d.Y).carrier
(d.Q.descendant d.X).measurable (d.Q.descendant d.Y).measurable
(fun x hx y hy => d.separated_distance_ratio hQ hs (z := (x,y)) hx hy)
d.levelMass d.levelMass _ _ _ (measurable_sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y)))
(d.inverseEndpointSymbol_measurable b.2 0) (d.inverseEndpointSymbol_measurable b.2 1)
(by norm_num : (0:ℝ) ≤ 1) hC hC (sourceConeCircle_norm_le (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y)))
(d.inverseEndpointSymbol_norm hQ j hb 0) (d.inverseEndpointSymbol_norm hQ j hb 1) r
simpa only [sourceParentDistanceF, inverseDistanceEndpointKernel, cellPair,
zpow_neg,zpow_ofNat,pow_two,inv_pow,Complex.ofReal_inv,Complex.ofReal_pow] using he
end PositiveDistanceData
end PlanarFalconer
end OAI
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