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67aee1d | 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.Analysis.FiniteFactor.Sampling.Kernel
namespace OAI
namespace FiniteFactor
open MeasureTheory Finset
noncomputable section
theorem samplingCharacter_nat (r : ℕ) (t : ℝ) :
samplingCharacter (r : ℤ) t = samplingCharacter 1 t ^ r := by
induction r with
| zero => simp [samplingCharacter_zero]
| succ r ih =>
rw [Nat.cast_add, Nat.cast_one, samplingCharacter_add, ih, pow_succ]
/-- Geometric-sum cancellation controls the kernel away from the origin. -/
theorem samplingKernel_geometric_bound (H : ℕ) (t : ℝ) :
‖∑ r ∈ Finset.range H, samplingCharacter (r : ℤ) t‖ *
‖samplingCharacter 1 t - 1‖ ≤ 2 := by
simp_rw [samplingCharacter_nat]
rw [← norm_mul, geom_sum_mul]
calc
_ ≤ ‖samplingCharacter 1 t ^ H‖ + ‖(1 : ℂ)‖ := norm_sub_le _ _
_ = _ := by norm_num [norm_pow, samplingCharacter_norm]
/-- Elementary chord lower bound, including both halves of the unit interval. -/
theorem samplingCharacter_chord_lower (δ t : ℝ) (hδ : 0 ≤ δ)
(ht : δ ≤ t) (ht' : t ≤ 1-δ) :
4*δ ≤ ‖samplingCharacter 1 t-1‖ := by
have he : samplingCharacter 1 t = Complex.exp (Complex.I * ((2*Real.pi*t : ℝ) : ℂ)) := by
unfold samplingCharacter
congr 1
push_cast
ring
rw [he, Complex.norm_exp_I_mul_ofReal_sub_one]
have harg : 2*Real.pi*t/2 = Real.pi*t := by ring
rw [harg, Real.norm_eq_abs, abs_mul, show |(2 : ℝ)| = 2 by norm_num]
have hsin : 2*δ ≤ Real.sin (Real.pi*t) := by
by_cases hhalf : t ≤ 1/2
· have ht0 : 0 ≤ t := hδ.trans ht
have hh := Real.mul_le_sin (x := Real.pi*t) (by positivity)
(by nlinarith [Real.pi_pos])
have he : 2/Real.pi*(Real.pi*t) = 2*t := by field_simp
rw [he] at hh
linarith
· have ht0 : 0 ≤ 1-t := by linarith
have hh := Real.mul_le_sin (x := Real.pi*(1-t)) (by positivity)
(by nlinarith [Real.pi_pos])
have he : 2/Real.pi*(Real.pi*(1-t)) = 2*(1-t) := by field_simp
rw [he, show Real.pi*(1-t) = Real.pi-Real.pi*t by ring, Real.sin_pi_sub] at hh
linarith
have := le_abs_self (Real.sin (Real.pi*t))
linarith
theorem samplingKernel_tail_bound (H : ℕ) (hH : 0 < H) (δ t : ℝ)
(hδ : 0 < δ) (ht : δ ≤ t) (ht' : t ≤ 1-δ) :
samplingKernel H t ≤ 1/(4*(H : ℝ)*δ^2) := by
have hc := samplingCharacter_chord_lower δ t hδ.le ht ht'
have hg := samplingKernel_geometric_bound H t
let A := ‖∑ r ∈ Finset.range H, samplingCharacter (r : ℤ) t‖
have hA : 0 ≤ A := norm_nonneg _
have hbound : A ≤ 1/(2*δ) := by
apply (le_div_iff₀ (by positivity : 0 < 2*δ)).mpr
have hh := mul_le_mul_of_nonneg_left hc hA
dsimp [A] at *
nlinarith
have hsq := pow_le_pow_left₀ hA hbound 2
have hHR : 0 < (H : ℝ) := by exact_mod_cast hH
unfold samplingKernel
calc
_ ≤ (1/(2*δ))^2/(H : ℝ) := div_le_div_of_nonneg_right hsq hHR.le
_ = _ := by ring
/-- Actual mass of the part of the Fejer kernel outside a delta-neighborhood
of the integer cut. -/
theorem samplingKernel_tail_integral (H : ℕ) (hH : 0 < H) (δ : ℝ) (hδ : 0 < δ) :
(∫ t in Set.Icc δ (1-δ), samplingKernel H t ∂unitIntervalMeasure) ≤
1/(4*(H : ℝ)*δ^2) := by
have hki : Integrable (samplingKernel H) unitIntervalMeasure :=
(samplingKernel_continuous H).integrableOn_Icc.mono_set Set.Ico_subset_Icc_self
rw [← integral_indicator measurableSet_Icc]
have hle : ∀ t, (Set.Icc δ (1-δ)).indicator (samplingKernel H) t ≤ 1/(4*(H : ℝ)*δ^2) := by
intro t
by_cases ht : t ∈ Set.Icc δ (1-δ)
· rw [Set.indicator_of_mem ht]
exact samplingKernel_tail_bound H hH δ t hδ ht.1 ht.2
· rw [Set.indicator_of_notMem ht]
positivity
have hh := integral_mono_ae (hki.indicator measurableSet_Icc) (integrable_const (1/(4*(H : ℝ)*δ^2)))
(Filter.Eventually.of_forall hle)
simpa using hh
end
end FiniteFactor
end OAI
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