| import OAI.Analysis.FiniteFactor.Sampling.Smoothing |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory |
| noncomputable section |
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| /-- Interval membership cannot change under a displacement smaller than |
| delta when the starting point is farther than delta from both endpoints. -/ |
| theorem interval_membership_stable (a b δ t u : ℝ) (_hδ : 0 ≤ δ) |
| (ha : δ < |t-a|) (hb : δ < |t-b|) (htu : |t-u| < δ) : |
| u ∈ Set.Icc a b ↔ t ∈ Set.Icc a b := by |
| have hd := abs_lt.mp htu |
| constructor |
| · intro hu |
| constructor |
| · by_contra ht |
| have hh : t-a < 0 := by linarith |
| rw [abs_of_neg hh] at ha |
| linarith [hu.1] |
| · by_contra ht |
| have hh : 0 < t-b := by linarith |
| rw [abs_of_pos hh] at hb |
| linarith [hu.2] |
| · intro ht |
| rw [abs_of_nonneg (sub_nonneg.mpr ht.1)] at ha |
| rw [abs_of_nonpos (sub_nonpos.mpr ht.2)] at hb |
| constructor <;> linarith |
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|
| /-- A separated pair of representatives, with the first away from the cut, |
| has a fractional displacement outside the kernel's exceptional arc. -/ |
| theorem fract_displacement_middle (δ t u : ℝ) (hδ : 0 < δ) |
| (ht : δ ≤ t) (ht' : t ≤ 1-δ) (hu : u ∈ Set.Ico (0 : ℝ) 1) |
| (hsep : δ ≤ |t-u|) : Int.fract (t-u) ∈ Set.Icc δ (1-δ) := by |
| by_cases hpos : 0 ≤ t-u |
| · rw [abs_of_nonneg hpos] at hsep |
| rw [Int.fract_eq_self.mpr ⟨hpos, by linarith [hu.1]⟩] |
| constructor <;> linarith [hu.1] |
| · have hneg : t-u < 0 := lt_of_not_ge hpos |
| rw [abs_of_neg hneg] at hsep |
| have hf : ⌊t-u⌋ = (-1 : ℤ) := Int.floor_eq_iff.mpr ⟨by norm_num; linarith [hu.2], by norm_num; linarith⟩ |
| rw [Int.fract, hf] |
| norm_num |
| constructor <;> linarith [hu.2] |
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|
| /-- The actual interval convolution differs from its indicator only by |
| the proved kernel tail, away from interval endpoints and the circle cut. -/ |
| theorem samplingSmooth_interval_error (H : ℕ) (hH : 0 < H) (a b δ t : ℝ) |
| (hδ : 0 < δ) (ht : δ ≤ t) (ht' : t ≤ 1-δ) |
| (ha : δ < |t-a|) (hb : δ < |t-b|) : |
| |samplingSmooth H (Set.Icc a b) t - (Set.Icc a b).indicator (fun _ => (1 : ℝ)) t| ≤ |
| 1/(4*(H : ℝ)*δ^2) := by |
| let E := Set.Icc a b |
| let K := fun u => samplingKernel H (t-u) |
| let C := 1/(4*(H : ℝ)*δ^2) |
| have hC : 0 ≤ C := by dsimp [C]; positivity |
| have hki : Integrable K unitIntervalMeasure := |
| ((samplingKernel_continuous H).comp (continuous_const.sub continuous_id)).integrableOn_Icc.mono_set |
| Set.Ico_subset_Icc_self |
| have hcut (B : Set ℝ) (hBm : MeasurableSet B) |
| (hB : ∀ u ∈ B, ¬(u ∈ E ↔ t ∈ E)) : |
| (∫ u in B, K u ∂unitIntervalMeasure) ≤ C := by |
| rw [← integral_indicator hBm] |
| have hle : ∀ᵐ u ∂unitIntervalMeasure, B.indicator K u ≤ C := by |
| filter_upwards [ae_restrict_mem measurableSet_Ico] with u hu |
| by_cases hub : u ∈ B |
| · rw [Set.indicator_of_mem hub] |
| have hsep : δ ≤ |t-u| := by |
| by_contra hh |
| exact hB u hub (interval_membership_stable a b δ t u hδ.le ha hb (lt_of_not_ge hh)) |
| have hf := fract_displacement_middle δ t u hδ ht ht' hu hsep |
| dsimp [K,C] |
| rw [← samplingKernel_fract H (t-u)] |
| exact samplingKernel_tail_bound H hH δ _ hδ hf.1 hf.2 |
| · rw [Set.indicator_of_notMem hub] |
| exact hC |
| have hh := integral_mono_ae (hki.indicator hBm) (integrable_const C) hle |
| simpa using hh |
| have hbounds := samplingSmooth_bounds H hH E t |
| by_cases hte : t ∈ E |
| · rw [Set.indicator_of_mem hte, abs_of_nonpos (by linarith [hbounds.2])] |
| have hh := hcut Eᶜ measurableSet_Icc.compl (fun u hu he => hu (he.mpr hte)) |
| have hsplit := integral_add_compl (s := E) measurableSet_Icc hki |
| have hone : (∫ u, K u ∂unitIntervalMeasure) = 1 := samplingKernel_shift_integral H hH t |
| change (∫ u in E, K u ∂unitIntervalMeasure) + (∫ u in Eᶜ, K u ∂unitIntervalMeasure) = _ at hsplit |
| change -(samplingSmooth H E t-1) ≤ C |
| have heq : (∫ u in E, K u ∂unitIntervalMeasure) = samplingSmooth H E t := rfl |
| rw [hone,heq] at hsplit |
| linarith |
| · rw [Set.indicator_of_notMem hte, sub_zero, abs_of_nonneg hbounds.1] |
| exact hcut E measurableSet_Icc (fun u hu he => hte (he.mp hu)) |
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| end |
| end FiniteFactor |
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| end OAI |
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