File size: 3,140 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 | import OAI.MeasureTheory.Falconer.Shells.HighShellSeparatedTerms
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
namespace PositiveDistanceData
theorem normalizedHighShellF_finite_separation (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (hs : d.StronglySeparated) (n : ℕ) (v r : ℝ) :
d.normalizedHighShellF n v r =
(((r*(2:ℝ)^(-(d.a:ℝ)))^(1/2:ℝ):ℝ):ℂ) * ∑ j ∈ Finset.range (n+1),
((((r/(2:ℝ)^v)⁻¹)^j:ℝ):ℂ) *
((inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)).map (fun b =>
d.absorbedInverseCoefficient j v b *
∫ 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)).sum := by
let L := fun j => inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)
let R := fun j => ((((r/(2:ℝ)^v)⁻¹)^j:ℝ):ℂ)
let A := (((r*(2:ℝ)^(-(d.a:ℝ)))^(1/2:ℝ):ℝ):ℂ)
let K := fun j (b : ℂ × (PairMaskIndex → ℕ)) => d.inverseDistanceEndpointKernel j b.2 r
let c := fun j b => d.absorbedInverseCoefficient j v b
have hi (j : ℕ) (b : ℂ × (PairMaskIndex → ℕ)) (hb : b ∈ L j) :
Integrable (fun z => c j b * K j b z) d.cellPair :=
(d.inverseDistanceEndpointKernel_integrable hQ hs j hb r).const_mul (c j b)
have hiJ (j : ℕ) : Integrable (fun z => R j * ((L j).map (fun b => c j b * K j b z)).sum) d.cellPair :=
(integrable_list_sum d.cellPair (L j) (fun b z => c j b * K j b z) (hi j)).const_mul (R j)
have heJ (j : ℕ) :
(∫ z, R j * ((L j).map (fun b => c j b * K j b z)).sum ∂d.cellPair) =
R j * ((L j).map (fun b => c j b * ∫ z, K j b z ∂d.cellPair)).sum := by
rw [integral_const_mul, integral_list_sum d.cellPair (L j) _ (hi j)]
simp_rw [integral_const_mul]
change (d.levelMass^(-2:ℤ):ℝ) • (∫ z, A * ∑ j ∈ Finset.range (n+1),
R j * ((L j).map (fun b => c j b * K j b z)).sum ∂d.cellPair) = _
rw [integral_const_mul,integral_finsetSum (Finset.range (n+1)) (fun j _ => hiJ j)]
simp_rw [heJ]
calc
_ = A * ∑ j ∈ Finset.range (n+1), R j * ((L j).map (fun b => c j b *
((d.levelMass^(-2:ℤ):ℝ) • ∫ z, K j b z ∂d.cellPair))).sum := by
simp only [Complex.real_smul, Finset.mul_sum, ← List.sum_map_mul_left]
apply Finset.sum_congr rfl
intro j hj
apply congrArg List.sum
apply List.map_congr_left
intro b hb
ring
_ = _ := by
congr 1
apply Finset.sum_congr rfl
intro j hj
congr 1
apply congrArg List.sum
apply List.map_congr_left
intro b hb
rw [show (d.levelMass^(-2:ℤ):ℝ) • (∫ z, K j b z ∂d.cellPair) = _ from
d.inverseDistanceEndpointKernel_form hQ hs j hb r]
end PositiveDistanceData
end PlanarFalconer
end OAI
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