| import OAI.Analysis.FiniteFactor.Sampling.KernelTail |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Finset |
| open scoped ComplexConjugate ENNReal |
| noncomputable section |
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| 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 |
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| 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 |
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| theorem samplingKernel_fract (H : ℕ) (t : ℝ) : |
| samplingKernel H (Int.fract t) = samplingKernel H t := by |
| simp only [samplingKernel, samplingCharacter_fract] |
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| 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 |
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| def samplingSmooth (H : ℕ) (E : Set ℝ) (t : ℝ) : ℝ := |
| ∫ u in E, samplingKernel H (t-u) ∂unitIntervalMeasure |
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| def samplingIntervalCoefficient (E : Set ℝ) (v : ℤ) : ℂ := |
| ∫ u in E, samplingCharacter (-v) u ∂unitIntervalMeasure |
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| 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 |
|
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| theorem samplingIntervalCoefficient_zero (E : Set ℝ) : |
| samplingIntervalCoefficient E 0 = ((unitIntervalMeasure E).toReal : ℂ) := by |
| simp [samplingIntervalCoefficient, samplingCharacter_zero, measureReal_def] |
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| 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 |
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| / |
| 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 |
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|
| end |
| end FiniteFactor |
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| end OAI |
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