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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 98 99 100 101 102 | import OAI.Analysis.FiniteFactor.Sampling.KernelTail
namespace OAI
namespace FiniteFactor
open MeasureTheory Finset
open scoped ComplexConjugate ENNReal
noncomputable section
theorem samplingCharacter_sub_argument (v : ℤ) (t u : ℝ) :
samplingCharacter v (t-u) = samplingCharacter v t * samplingCharacter (-v) u := by
unfold samplingCharacter
rw [← Complex.exp_add]
congr 1
push_cast
ring
theorem samplingCharacter_fract (v : ℤ) (t : ℝ) :
samplingCharacter v (Int.fract t) = samplingCharacter v t := by
simpa only [samplingCharacter, mul_assoc] using fourier_fract v t
theorem samplingKernel_fract (H : ℕ) (t : ℝ) :
samplingKernel H (Int.fract t) = samplingKernel H t := by
simp only [samplingKernel, samplingCharacter_fract]
theorem samplingKernel_shift_integral (H : ℕ) (hH : 0 < H) (t : ℝ) :
(∫ u, samplingKernel H (t-u) ∂unitIntervalMeasure) = 1 := by
have hc : (∫ u, (samplingKernel H (t-u) : ℂ) ∂unitIntervalMeasure) = 1 := by
simp_rw [samplingKernel_expansion, samplingCharacter_sub_argument]
rw [integral_const_mul, integral_finsetSum _ (fun r _ =>
integrable_finsetSum _ (fun q _ => (samplingCharacter_integrable _).const_mul _))]
simp_rw [integral_finsetSum _ (fun q _ => (samplingCharacter_integrable _).const_mul _),
integral_const_mul, samplingCharacter_integral]
simp [sub_eq_zero, -Finset.mem_range, hH.ne', samplingCharacter_zero]
rw [integral_complex_ofReal] at hc
exact_mod_cast hc
def samplingSmooth (H : ℕ) (E : Set ℝ) (t : ℝ) : ℝ :=
∫ u in E, samplingKernel H (t-u) ∂unitIntervalMeasure
def samplingIntervalCoefficient (E : Set ℝ) (v : ℤ) : ℂ :=
∫ u in E, samplingCharacter (-v) u ∂unitIntervalMeasure
theorem samplingSmooth_bounds (H : ℕ) (hH : 0 < H) (E : Set ℝ) (t : ℝ) :
0 ≤ samplingSmooth H E t ∧ samplingSmooth H E t ≤ 1 := by
have hki : Integrable (fun u => samplingKernel H (t-u)) unitIntervalMeasure :=
((samplingKernel_continuous H).comp (continuous_const.sub continuous_id)).integrableOn_Icc.mono_set
Set.Ico_subset_Icc_self
constructor
· exact integral_nonneg (fun u => samplingKernel_nonneg H (t-u))
· calc
_ ≤ ∫ u, samplingKernel H (t-u) ∂unitIntervalMeasure :=
integral_mono_measure Measure.restrict_le_self
(Filter.Eventually.of_forall (fun u => samplingKernel_nonneg H (t-u))) hki
_ = 1 := samplingKernel_shift_integral H hH t
theorem samplingIntervalCoefficient_zero (E : Set ℝ) :
samplingIntervalCoefficient E 0 = ((unitIntervalMeasure E).toReal : ℂ) := by
simp [samplingIntervalCoefficient, samplingCharacter_zero, measureReal_def]
theorem samplingIntervalCoefficient_norm (E : Set ℝ) (v : ℤ) :
‖samplingIntervalCoefficient E v‖ ≤ 1 := by
unfold samplingIntervalCoefficient
calc
_ ≤ ∫ u in E, ‖samplingCharacter (-v) u‖ ∂unitIntervalMeasure := norm_integral_le_integral_norm _
_ = (unitIntervalMeasure E).toReal := by simp [samplingCharacter_norm, measureReal_def]
_ ≤ 1 := by
have hh : unitIntervalMeasure E ≤ 1 :=
(measure_mono (Set.subset_univ E)).trans_eq (measure_univ)
simpa using ENNReal.toReal_mono (by norm_num : (1 : ℝ≥0∞) ≠ ∞) hh
/-- An actual finite Fourier expansion of the smoothed interval indicator.
Keeping the two kernel indices avoids assuming a coefficient formula. -/
theorem samplingSmooth_expansion (H : ℕ) (E : Set ℝ) (t : ℝ) :
(samplingSmooth H E t : ℂ) = (H : ℂ)⁻¹ *
∑ r ∈ Finset.range H, ∑ q ∈ Finset.range H,
samplingCharacter ((r : ℤ)-(q : ℤ)) t *
samplingIntervalCoefficient E ((r : ℤ)-(q : ℤ)) := by
rw [samplingSmooth, ← integral_complex_ofReal]
simp_rw [samplingKernel_expansion, samplingCharacter_sub_argument]
rw [integral_const_mul, integral_finsetSum _ (fun r _ =>
integrable_finsetSum _ (fun q _ => ((samplingCharacter_integrable _).restrict).const_mul _))]
simp_rw [integral_finsetSum _ (fun q _ => ((samplingCharacter_integrable _).restrict).const_mul _),
integral_const_mul]
rfl
theorem samplingSmooth_continuous (H : ℕ) (E : Set ℝ) : Continuous (samplingSmooth H E) := by
have hc : Continuous (fun t => (samplingSmooth H E t : ℂ)) := by
simp_rw [samplingSmooth_expansion]
apply Continuous.const_mul
apply continuous_finsetSum
intro r hr
apply continuous_finsetSum
intro q hq
exact (samplingCharacter_continuous _).mul_const _
simpa only [Function.comp_def, Complex.ofReal_re] using Complex.continuous_re.comp hc
end
end FiniteFactor
end OAI
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