| import OAI.MeasureTheory.Falconer.Shells.NormalizedHighShellError | |
| import OAI.MeasureTheory.Falconer.Fourier.WeightedDistanceFourier | |
| namespace OAI | |
| namespace PlanarFalconer | |
| open Set MeasureTheory Filter | |
| namespace PositiveDistanceData | |
| noncomputable def normalizedHighShellF (d : PositiveDistanceData) (n : ℕ) (v r : ℝ) : ℂ := | |
| (d.levelMass^(-2:ℤ):ℝ) • ∫ z, d.normalizedHighShellKernel n v r | |
| (sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y))) z ∂d.cellPair | |
| theorem normalizedHighShellF_error_of_pointwise (d : PositiveDistanceData) [IsFiniteMeasure d.ρ] | |
| (hQ : 0 < d.Q.side) (hs : d.StronglySeparated) (n : ℕ) (v r ε : ℝ) | |
| (herr : ∀ x ∈ (d.Q.descendant d.X).carrier, ∀ y ∈ (d.Q.descendant d.Y).carrier, | |
| ‖(d.spatialWeight (x,y):ℂ)*Complex.exp (Complex.I*((r*dist x y:ℝ):ℂ)) - | |
| d.normalizedHighShellKernel n v r | |
| (sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y))) (x,y)‖ ≤ ε) : | |
| Integrable (d.normalizedHighShellKernel n v r | |
| (sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y)))) d.cellPair ∧ | |
| ‖charFun d.weightedDistance r - d.normalizedHighShellF n v r‖ ≤ | |
| d.levelMass^(-2:ℤ) * d.cellPair.real univ * ε := by | |
| let ψ := sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y)) | |
| let K := d.normalizedHighShellKernel n v r ψ | |
| let f : Plane × Plane → ℂ := fun z => | |
| (d.spatialWeight z:ℂ)*Complex.exp (Complex.I*((r*dist z.1 z.2:ℝ):ℂ)) | |
| have hfm : Measurable f := by | |
| apply Measurable.mul (Complex.measurable_ofReal.comp d.measurable_spatialWeight) | |
| fun_prop | |
| have hfb : ∀ᵐ z ∂d.cellPair, ‖f z‖ ≤ 1 := by | |
| filter_upwards [d.ae_cellPair_mem] with z hz | |
| have hw₀ := d.spatialWeight_nonneg hQ z | |
| have hw₁ := d.spatialWeight_le_one hQ hs hz.1 hz.2 | |
| simpa [f,norm_mul,Complex.norm_exp,Complex.mul_re,Complex.mul_im, | |
| Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hw₀] using hw₁ | |
| have hfi : Integrable f d.cellPair := Integrable.of_bound hfm.aestronglyMeasurable 1 hfb | |
| have hkm : Measurable K := (d.normalizedHighShellKernel_measurable n v ψ | |
| (measurable_sourceConeCircle _)).comp (f := fun z : Plane × Plane => (r,z)) | |
| (measurable_const.prodMk measurable_id) | |
| have he : ∀ᵐ z ∂d.cellPair, ‖f z-K z‖ ≤ ε := by | |
| filter_upwards [d.ae_cellPair_mem] with z hz | |
| exact herr z.1 hz.1 z.2 hz.2 | |
| have hki : Integrable K d.cellPair := by | |
| refine Integrable.of_bound hkm.aestronglyMeasurable (ε+1) ?_ | |
| filter_upwards [he,hfb] with z hz hb | |
| calc | |
| ‖K z‖ = ‖(K z-f z)+f z‖ := by rw [sub_add_cancel] | |
| _ ≤ ‖K z-f z‖+‖f z‖ := norm_add_le _ _ | |
| _ ≤ ε+1 := add_le_add (by simpa only [norm_sub_rev] using hz) hb | |
| refine ⟨hki,?_⟩ | |
| have hfour : charFun d.weightedDistance r = (d.levelMass^(-2:ℤ):ℝ) • ∫ z, f z ∂d.cellPair := by | |
| rw [d.weightedDistance_fourier hQ] | |
| simp only [f,mul_comm Complex.I] | |
| rw [hfour,normalizedHighShellF,← smul_sub] | |
| change ‖(d.levelMass^(-2:ℤ):ℝ) • ((∫ z, f z ∂d.cellPair)-(∫ z, K z ∂d.cellPair))‖ ≤ _ | |
| rw [← integral_sub hfi hki,norm_smul,Real.norm_eq_abs, | |
| abs_of_nonneg (by positivity : 0 ≤ d.levelMass^(-2:ℤ))] | |
| have hb := norm_integral_le_of_norm_le_const he | |
| have hh := mul_le_mul_of_nonneg_left hb (by positivity : 0 ≤ d.levelMass^(-2:ℤ)) | |
| exact hh.trans_eq (by ring) | |
| theorem uniform_high_shell_fourier_error {sig T : ℝ} (hsig : 0 < sig) : | |
| ∃ n N₀ : ℕ, ∀ d : PositiveDistanceData, ∀ [IsFiniteMeasure d.ρ], | |
| N₀ ≤ d.N → d.sig = sig → 0 < d.Q.side → d.StronglySeparated → 0 < d.q → | |
| d.sig*d.N ≤ (d.q:ℝ) → d.a+100*d.q < d.t → d.t ≤ d.N → | |
| ∀ v r : ℝ, (d.t:ℝ)-d.sig*d.N ≤ v → (2:ℝ)^(v-1) ≤ r → | |
| ‖charFun d.weightedDistance r - d.normalizedHighShellF n v r‖ ≤ | |
| d.levelMass^(-2:ℤ) * d.cellPair.real univ * ((2:ℝ)^d.N)^(-T) := by | |
| obtain ⟨n,N₀,h₀⟩ := uniform_normalized_high_shell_error (T := T) hsig | |
| refine ⟨n,N₀,?_⟩ | |
| intro d inst hN hd hQ hs hq hqs hw ht v r hv hr | |
| exact (d.normalizedHighShellF_error_of_pointwise hQ hs n v r _ | |
| (h₀ d hN hd hQ hs hq hqs hw ht v r hv hr)).2 | |
| end PositiveDistanceData | |
| end PlanarFalconer | |
| end OAI | |