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d4317d7 | 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 | import OAI.MeasureTheory.Falconer.Estimates.SelectedDistanceAssembly
import OAI.MeasureTheory.Falconer.Graphs.GraphOriginalNonedgePower
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
open scoped ComplexConjugate
theorem distanceBinIntegrand_cross (ρ ν : Measure Plane) (X Y X' Y' : Set Plane)
(M₁ M₂ : ℝ) (b₁ b₂ : Plane → UnitCircle → ℂ) (axis : ℝ) (n j k : ℕ)
(r : ℝ) (u w : UnitCircle) :
distanceBinIntegrand ρ ν X Y M₁ M₂ b₁ b₂ axis n j (r,u) *
conj (distanceBinIntegrand ρ ν X' Y' M₁ M₂ b₁ b₂ axis n k (r,w)) =
sourceGraphFourTransform ![ρ,ν,ν,ρ] ![X,Y,Y',X'] ![M₁,M₂,M₂,M₁] ![b₁,b₂,b₂,b₁]
(graphSourceWeight true true axis n j k) r u w := by
simp only [sourceGraphFourTransform,graphSourceWeight,↓reduceIte,graphRotatedDistanceWeight_eq,
Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_two,Matrix.cons_val_three,
Matrix.head_cons,Matrix.tail_cons,distanceBinIntegrand,map_mul,starRingEnd_self_apply]
ring
theorem graphOriginalTerm_profile_normalization
(ρ : Fin 4 → Measure Plane) (X : Fin 4 → Set Plane) (Mp Ma : Fin 4 → ℝ)
(hM : ∀ i, Mp i ≠ 0) (b : Fin 4 → Plane → UnitCircle → ℂ)
(distance plus : Bool) (axis δ v : ℝ) (j k : ℕ) (Ψ : ℝ → ℝ) :
graphOriginalTerm ρ X Ma b distance plus axis δ v j k Ψ =
((Mp 0/Ma 0)*(Mp 1/Ma 1)*(Mp 2/Ma 2)*(Mp 3/Ma 3)) •
graphOriginalTerm ρ X Mp b distance plus axis δ v j k Ψ := by
unfold graphOriginalTerm
simp_rw [sourceGraphFourTransform_profile_normalization ρ X Mp Ma hM b,
smul_comm (Ψ _) ((Mp 0/Ma 0)*(Mp 1/Ma 1)*(Mp 2/Ma 2)*(Mp 3/Ma 3)), integral_smul]
exact smul_comm _ _ _
end PlanarFalconer
end OAI
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