| import OAI.MeasureTheory.Falconer.Angular.SphereInverse |
| import Mathlib.Analysis.Normed.Module.Normalize |
|
|
| namespace OAI |
|
|
| /-! # Inverse stationary phase for the original Euclidean displacement |
|
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| The direction and oscillation parameter are exactly `z/‖z‖` and `r*‖z‖`. |
| No projected distance is substituted for the original distance. |
| -/ |
|
|
| namespace Falconer |
| open Set Metric MeasureTheory NormedSpace |
| open scoped ContDiff |
|
|
| noncomputable def displacementDirection {d : ℕ} (z : Space d) (hz : z ≠ 0) : Sphere d := |
| ⟨normalize z,by rw [mem_sphere,dist_zero_right,norm_normalize hz]⟩ |
|
|
| theorem sphereOscillation_displacement {d : ℕ} (r : ℝ) (z : Space d) (hz : z ≠ 0) |
| (B : Sphere d → ℂ) : |
| sphereOscillation (r*‖z‖) (displacementDirection z hz) B = |
| ∫ w : Sphere d, Complex.exp (Complex.I*(r : ℂ)*(inner ℝ (w : Space d) z : ℂ))*B w |
| ∂sphereMeasure d := by |
| unfold sphereOscillation |
| apply integral_congr_ae |
| filter_upwards with w |
| congr 2 |
| have hi : (r*‖z‖)*inner ℝ (w : Space d) (normalize z) = r*inner ℝ (w : Space d) z := by |
| rw [mul_assoc,← real_inner_smul_right,norm_smul_normalize] |
| have hc := congrArg (fun a : ℝ => (a : ℂ)) hi |
| push_cast at hc |
| push_cast |
| change Complex.I*((r : ℂ)*(‖z‖ : ℂ))*(inner ℝ (w : Space d) (normalize z) : ℂ) = _ |
| calc |
| _ = Complex.I*(((r : ℂ)*(‖z‖ : ℂ))*(inner ℝ (w : Space d) (normalize z) : ℂ)) := by ring |
| _ = Complex.I*((r : ℂ)*(inner ℝ (w : Space d) z : ℂ)) := congrArg (fun a => Complex.I*a) hc |
| _ = _ := by ring |
|
|
| noncomputable def inverseDisplacementSum (m J : ℕ) (r : ℝ) (z : Space (m+3)) |
| (B : Space (m+3) → ℂ) : ℂ := |
| ∑ j ∈ Finset.range (J+1), (((r*‖z‖)^(-(j : ℝ)) : ℝ) : ℂ)* |
| ∫ w : Sphere (m+3), Complex.exp (Complex.I*(r : ℂ)*(inner ℝ (w : Space (m+3)) z : ℂ))* |
| localInverseAmplitude m B j w ∂sphereMeasure (m+3) |
|
|
| theorem inverseSphereSum_displacement (m J : ℕ) (r : ℝ) (z : Space (m+3)) (hz : z ≠ 0) |
| (B : Space (m+3) → ℂ) : |
| inverseSphereSum m J (r*‖z‖) B (displacementDirection z hz) = |
| inverseDisplacementSum m J r z B := by |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [sphereOscillation_displacement] |
|
|
| theorem displacement_inverse_expansion |
| (hgnomonic : PublishedGnomonicSurface) (hquadratic : PublishedQuadraticStationary) |
| (m J : ℕ) : |
| ∃ A : ℝ, 0 ≤ A ∧ ∀ (z : Space (m+3)), z ≠ 0 → |
| ‖normalize z-(gnomonicChart (0 : Space (m+2)) : Space (m+3))‖ ≤ 1/8 → |
| ∀ B : Space (m+3) → ℂ, ContDiff ℝ ∞ B → |
| ∀ C D : ℝ, 0 ≤ C → 1 ≤ D → |
| (∀ j ≤ 2*(J+1)+(m+2)+1, ∀ w : Sphere (m+3), |
| ‖iteratedFDeriv ℝ j B w‖ ≤ C*D^j) → |
| ∀ r : ℝ, 1 ≤ r*‖z‖ → |
| ‖(((r*‖z‖)^(((m+2 : ℕ) : ℝ)/2) : ℝ) : ℂ)*inverseDisplacementSum m J r z B - |
| Complex.exp (Complex.I*((r*‖z‖ : ℝ) : ℂ))*B (normalize z)‖ ≤ |
| A*(r*‖z‖)^(-((J+1 : ℕ) : ℝ))*C*D^(2*(J+1)+(m+2)+1) := by |
| obtain ⟨A,hA,hAb⟩ := sphere_inverse_expansion hgnomonic hquadratic m J |
| refine ⟨A,hA,?_⟩ |
| intro z hz he B hB C D hC hD hb r hr |
| have hh := hAb (displacementDirection z hz) he B hB C D hC hD hb (r*‖z‖) hr |
| rw [inverseSphereSum_displacement] at hh |
| exact hh |
|
|
| end Falconer |
|
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| end OAI |
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|
