File size: 4,353 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 77 78 79 80 81 | 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
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