| import OAI.MeasureTheory.Falconer.Angular.CompactStationaryAmplitude |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open MeasureTheory Set |
| open scoped SchwartzMap ContDiff |
|
|
| theorem compact_quadratic_stationary_expansion {b a z C D : ℝ} (hb : 0 < b) |
| (haz : a ≤ z) (φ : 𝓢(ℝ,ℂ)) (hs : tsupport (φ : ℝ → ℂ) ⊆ Icc a z) (n : ℕ) |
| (h₀ : ∀ x, ‖iteratedDeriv (2*(n+1)) (φ : ℝ → ℂ) x‖ ≤ C*D^(2*(n+1))) |
| (h₂ : ∀ x, ‖iteratedDeriv (2*(n+1)+2) (φ : ℝ → ℂ) x‖ ≤ C*D^(2*(n+1)+2)) : |
| ‖(∫ x : ℝ, Complex.exp (-((b:ℂ)*Complex.I)*(x:ℂ)^2)*φ x) - |
| fresnelCoefficient b * quadraticExpansion φ n b‖ ≤ |
| ((Real.pi/b)^(1/2:ℝ) * ((Real.pi^2/b)^(n+1)/(n.factorial:ℝ))) * |
| ((C*(z-a)*D^(2*(n+1))/(2*Real.pi)^(2*(n+1)) + |
| C*(z-a)*D^(2*(n+1)+2)/(2*Real.pi)^(2*(n+1)+2)) * |
| ∫ ξ : ℝ, (1+ξ^2)⁻¹) := by |
| apply (quadratic_stationary_expansion_L1 hb φ n).trans |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply mul_le_mul_of_nonneg_right _ (integral_nonneg (fun _ => by positivity)) |
| exact add_le_add (compact_fourierDerivativeL1_bound φ haz hs _ h₀) |
| (compact_fourierDerivativeL1_bound φ haz hs _ h₂) |
|
|
| theorem compact_quadratic_derivative_scale {a z C D : ℝ} (haz : a ≤ z) |
| (hC : 0 ≤ C) (hD : 1 ≤ D) (n : ℕ) : |
| C*(z-a)*D^(2*(n+1))/(2*Real.pi)^(2*(n+1)) + |
| C*(z-a)*D^(2*(n+1)+2)/(2*Real.pi)^(2*(n+1)+2) ≤ |
| (C*(z-a)*((2*Real.pi)^(2*(n+1)))⁻¹ + |
| C*(z-a)*((2*Real.pi)^(2*(n+1)+2))⁻¹) * D^(2*(n+1)+2) := by |
| have hp : D^(2*(n+1)) ≤ D^(2*(n+1)+2) := pow_le_pow_right₀ hD (by omega) |
| calc |
| _ ≤ C*(z-a)*D^(2*(n+1)+2)/(2*Real.pi)^(2*(n+1)) + |
| C*(z-a)*D^(2*(n+1)+2)/(2*Real.pi)^(2*(n+1)+2) := by |
| exact add_le_add |
| (div_le_div_of_nonneg_right |
| (mul_le_mul_of_nonneg_left hp (mul_nonneg hC (sub_nonneg.mpr haz))) |
| (show 0 ≤ (2 * Real.pi) ^ (2 * (n + 1)) by positivity)) le_rfl |
| _ = _ := by ring |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
