| import OAI.MeasureTheory.Falconer.Shells.HighShellSeparatedTerms |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set MeasureTheory |
| namespace PositiveDistanceData |
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| 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] |
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| end PositiveDistanceData |
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| end PlanarFalconer |
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| end OAI |
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