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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 99 100 101 102 103 104 105 | import OAI.Analysis.FiniteFactor.Sampling.Fourier
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
import Mathlib.Algebra.Ring.GeomSum
namespace OAI
namespace FiniteFactor
open MeasureTheory Finset
open scoped ComplexConjugate
noncomputable section
def samplingCharacter (v : ℤ) (t : ℝ) : ℂ :=
Complex.exp (2 * Real.pi * Complex.I * v * t)
theorem samplingCharacter_norm (v : ℤ) (t : ℝ) : ‖samplingCharacter v t‖ = 1 := by
simp [samplingCharacter, Complex.norm_exp, Complex.mul_re, Complex.mul_im]
theorem samplingCharacter_continuous (v : ℤ) : Continuous (samplingCharacter v) := by
unfold samplingCharacter
fun_prop
theorem samplingCharacter_zero (t : ℝ) : samplingCharacter 0 t = 1 := by simp [samplingCharacter]
theorem samplingCharacter_add (v w : ℤ) (t : ℝ) :
samplingCharacter (v+w) t = samplingCharacter v t * samplingCharacter w t := by
unfold samplingCharacter
rw [← Complex.exp_add]
congr 1
push_cast
ring
theorem samplingCharacter_neg (v : ℤ) (t : ℝ) :
samplingCharacter (-v) t = conj (samplingCharacter v t) := by
have htwo : conj (2 : ℂ) = 2 := by
change conj ((2 : ℝ) : ℂ) = _
exact Complex.conj_ofReal _
simp [samplingCharacter, ← Complex.exp_conj, mul_assoc, htwo]
theorem samplingCharacter_sub (v w : ℤ) (t : ℝ) :
samplingCharacter (v-w) t = samplingCharacter v t * conj (samplingCharacter w t) := by
rw [sub_eq_add_neg, samplingCharacter_add, samplingCharacter_neg]
theorem samplingCharacter_integrable (v : ℤ) :
Integrable (samplingCharacter v) unitIntervalMeasure :=
(samplingCharacter_continuous v).integrableOn_Icc.mono_set Set.Ico_subset_Icc_self
theorem samplingCharacter_integral (v : ℤ) :
(∫ t, samplingCharacter v t ∂unitIntervalMeasure) = if v = 0 then 1 else 0 := by
by_cases hv : v = 0
· subst v; simp [samplingCharacter_zero]
rw [ite_eq_right hv]
unfold unitIntervalMeasure
rw [restrict_Ico_eq_restrict_Ioc, ← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
change (∫ t in (0 : ℝ)..1, Complex.exp ((2 * Real.pi * Complex.I * (v : ℂ)) * t)) = 0
have hvC : (v : ℂ) ≠ 0 := by exact_mod_cast hv
have hcoef : 2 * (Real.pi : ℂ) * Complex.I * (v : ℂ) ≠ 0 :=
mul_ne_zero (mul_ne_zero (mul_ne_zero (by norm_num) (Complex.ofReal_ne_zero.mpr Real.pi_ne_zero))
Complex.I_ne_zero) hvC
rw [integral_exp_mul_complex hcoef]
simp only [Complex.ofReal_one, mul_one, Complex.ofReal_zero, mul_zero, Complex.exp_zero]
rw [show 2 * (Real.pi : ℂ) * Complex.I * (v : ℂ) = (v : ℂ)*(2*Real.pi*Complex.I) by ring,
Complex.exp_int_mul_two_pi_mul_I]
simp
/-- The finite Fejer kernel, as a nonnegative real function. -/
def samplingKernel (H : ℕ) (t : ℝ) : ℝ :=
‖∑ r ∈ Finset.range H, samplingCharacter (r : ℤ) t‖^2 / (H : ℝ)
theorem samplingKernel_nonneg (H : ℕ) (t : ℝ) : 0 ≤ samplingKernel H t := by
unfold samplingKernel
positivity
theorem samplingKernel_continuous (H : ℕ) : Continuous (samplingKernel H) := by
unfold samplingKernel
apply Continuous.div_const
apply Continuous.pow
apply Continuous.norm
exact continuous_finsetSum _ (fun r _ => samplingCharacter_continuous _)
theorem samplingKernel_expansion (H : ℕ) (t : ℝ) :
(samplingKernel H t : ℂ) = (H : ℂ)⁻¹ *
∑ r ∈ Finset.range H, ∑ q ∈ Finset.range H, samplingCharacter ((r : ℤ)-(q : ℤ)) t := by
simp only [samplingKernel, Complex.ofReal_div, Complex.ofReal_pow, Complex.ofReal_natCast,
samplingCharacter_sub, ← Finset.mul_sum, ← Finset.sum_mul, ← map_sum]
rw [← Complex.ofReal_pow, Complex.sq_norm, Complex.mul_conj]
ring
theorem samplingKernel_integral (H : ℕ) (hH : 0 < H) :
(∫ t, samplingKernel H t ∂unitIntervalMeasure) = 1 := by
have hki : Integrable (samplingKernel H) unitIntervalMeasure :=
(samplingKernel_continuous H).integrableOn_Icc.mono_set Set.Ico_subset_Icc_self
have hc : (∫ t, (samplingKernel H t : ℂ) ∂unitIntervalMeasure) = 1 := by
simp_rw [samplingKernel_expansion]
rw [integral_const_mul, integral_finsetSum _ (fun r _ =>
integrable_finsetSum _ (fun q _ => samplingCharacter_integrable _))]
simp_rw [integral_finsetSum _ (fun q _ => samplingCharacter_integrable _), samplingCharacter_integral]
simp [sub_eq_zero, -Finset.mem_range, hH.ne']
rw [integral_complex_ofReal] at hc
exact_mod_cast hc
end
end FiniteFactor
end OAI
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