File size: 3,334 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 | import OAI.MeasureTheory.Falconer.Angular.SphereInverse
import Mathlib.Analysis.Normed.Module.Normalize
namespace OAI
/-! # Inverse stationary phase for the original Euclidean displacement
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
end OAI
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