| import OAI.Geometry.Kahler.BaseDensityLowMoments |
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| namespace OAI |
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| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open Set Filter Topology |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff |
| noncomputable section |
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| open Set Filter Topology MeasureTheory |
| namespace PinchedHartogs.BaseConstruction |
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| lemma circle_real_char_mean {n : ℤ} (hn : n ≠ 0) : |
| (∫ z, (phaseChar n z).re ∂circleMeasure)=0 := by |
| have he : (∫ z, (phaseChar n z).re ∂circleMeasure) = (∫ z, phaseChar n z ∂circleMeasure).re := integral_re (compact_continuous_integrable (phaseChar_continuous n)) |
| rw [he,circle_char_mean,ite_eq_right hn] |
| rfl |
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| lemma circle_real_char_cos {n : ℕ} (hn : 0 < n) : |
| (∫ z : Circle, (phaseChar n z).re*(z:ℂ).re ∂circleMeasure)=if n=1 then 1/2 else 0 := by |
| obtain ⟨index, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hn) |
| have hh := phase_real_moment (1:ℂ) (1:ℤ) (index+1) (by norm_num) |
| have he : ∫ z : Circle, (phaseChar (index+1) z*((z:ℂ).re:ℂ)) ∂circleMeasure = if index+1=1 then (1/2:ℂ) else 0 := by |
| simpa [phaseChar,eq_comm] using hh |
| have hh' := congrArg Complex.re he |
| have hei : (∫ z : Circle, (phaseChar (index+1) z*((z:ℂ).re:ℂ)).re ∂circleMeasure) = (∫ z : Circle, phaseChar (index+1) z*((z:ℂ).re:ℂ) ∂circleMeasure).re := integral_re (compact_continuous_integrable ((phaseChar_continuous _).mul (Complex.continuous_ofReal.comp (Complex.continuous_re.comp continuous_subtype_val)))) |
| rw [← hei] at hh' |
| simpa only [Nat.succ_eq_add_one,Nat.cast_add,Nat.cast_one,apply_ite,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,mul_zero,sub_zero,Complex.zero_re,Complex.div_ofNat_re,Complex.one_re] using hh' |
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| lemma hasSum_circle_log {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) (z : Circle) : |
| HasSum (fun n : ℕ => (-2*r^(n+1)/(n+1))*(phaseChar (n+1) z).re) |
| (Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2)) := by |
| have hnorm : ‖(r:ℂ)*(z:ℂ)‖ < 1 := by rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hr,z.norm_coe,mul_one]; exact hr1 |
| have hh := (Complex.hasSum_re (Complex.hasSum_taylorSeries_neg_log' hnorm)).mul_left (-2) |
| have heq : -2 * (-Complex.log (1-(r:ℂ)*(z:ℂ))).re = Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2) := by |
| rw [Complex.neg_re,Complex.log_re,Real.log_pow] |
| ring |
| rw [heq] at hh |
| apply hh.congr_fun |
| intro n |
| have he : ((r:ℂ)*(z:ℂ))^(n+1)/((n:ℂ)+1) = |
| ((r^(n+1)/(n+1):ℝ):ℂ)*phaseChar ((n:ℤ)+1) z := by |
| have hz : phaseChar ((n:ℤ)+1) z=(z:ℂ)^(n+1) := by |
| rw [show (n:ℤ)+1=((n+1:ℕ):ℤ) by omega] |
| simp only [phaseChar,zpow_natCast,Circle.coe_pow] |
| rw [hz] |
| push_cast |
| rw [mul_pow] |
| ring |
| rw [he] |
| simp only [Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] |
| ring |
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|
| lemma circle_log_moments {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) : |
| (∫ z : Circle, Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2) ∂circleMeasure)=0 ∧ |
| (∫ z : Circle, Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2)*(z:ℂ).re ∂circleMeasure)=-r := by |
| let T : ℕ → Circle → ℝ := fun n z => (-2*r^(n+1)/(n+1))*(phaseChar (n+1) z).re |
| have hb : ∀ n z, |T n z| ≤ 2*r^(n+1) := by |
| intro n z |
| have hz : |(phaseChar (n+1) z).re| ≤ 1 := (Complex.abs_re_le_norm _).trans_eq (phaseChar_norm _ _) |
| have hn : (1:ℝ) ≤ n+1 := by have := Nat.cast_nonneg (α:=ℝ) n; linarith |
| calc |
| _ = (2*r^(n+1)/(n+1))*|(phaseChar (n+1) z).re| := by dsimp [T]; rw [abs_mul,abs_div,abs_mul,abs_pow,abs_of_nonneg hr,abs_of_pos (by positivity : (0:ℝ)<n+1)]; norm_num |
| _ ≤ 2*r^(n+1)/(n+1) := by nlinarith [div_nonneg (by positivity : 0 ≤ 2*r^(n+1)) (by positivity : (0:ℝ) ≤ n+1)] |
| _ ≤ 2*r^(n+1) := div_le_self (by positivity) hn |
| have hsum : Summable (fun n : ℕ => 2*r^(n+1)) := by |
| simpa only [pow_succ,mul_assoc,mul_left_comm,mul_comm] using |
| (summable_geometric_of_lt_one hr hr1).mul_left (2*r) |
| have hzre : ∀ z : Circle, |(z:ℂ).re| ≤ 1 := fun z => (Complex.abs_re_le_norm _).trans_eq z.norm_coe |
| have hmean : HasSum (fun n => ∫ z, T n z ∂circleMeasure) (∫ z : Circle, Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2) ∂circleMeasure) := by |
| apply hasSum_integral_of_dominated_convergence (fun n _ => 2*r^(n+1)) |
| · intro n; exact (continuous_const.mul (Complex.continuous_re.comp (phaseChar_continuous _))).aestronglyMeasurable |
| · intro n; exact Eventually.of_forall (fun z => by simpa only [Real.norm_eq_abs] using hb n z) |
| · exact Eventually.of_forall (fun _ => hsum) |
| · exact integrable_const _ |
| · exact Eventually.of_forall (hasSum_circle_log hr hr1) |
| have hcos : HasSum (fun n => ∫ z, T n z*(z:ℂ).re ∂circleMeasure) (∫ z : Circle, Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2)*(z:ℂ).re ∂circleMeasure) := by |
| apply hasSum_integral_of_dominated_convergence (fun n _ => 2*r^(n+1)) |
| · intro n; exact ((continuous_const.mul (Complex.continuous_re.comp (phaseChar_continuous _))).mul (Complex.continuous_re.comp continuous_subtype_val)).aestronglyMeasurable |
| · intro n; exact Eventually.of_forall (fun z => by |
| rw [Real.norm_eq_abs,abs_mul] |
| calc _ ≤ (2*r^(n+1))*1 := mul_le_mul (hb n z) (hzre z) (abs_nonneg _) (by positivity) |
| _ = _ := mul_one _) |
| · exact Eventually.of_forall (fun _ => hsum) |
| · exact integrable_const _ |
| · exact Eventually.of_forall (fun z => (hasSum_circle_log hr hr1 z).mul_right _) |
| have hm0 : ∀ n, (∫ z, T n z ∂circleMeasure)=0 := by |
| intro n |
| dsimp only [T] |
| rw [integral_const_mul,circle_real_char_mean (by omega : (n:ℤ)+1 ≠ 0),mul_zero] |
| have hc0 : ∀ n, (∫ z : Circle, T n z*(z:ℂ).re ∂circleMeasure)=if n=0 then -r else 0 := by |
| intro n |
| simp only [T,mul_assoc] |
| rw [integral_const_mul] |
| have hh := circle_real_char_cos (Nat.succ_pos n) |
| simp only [Nat.succ_eq_add_one, Nat.cast_add,Nat.cast_one] at hh |
| rw [hh] |
| by_cases hn : n=0 |
| · subst n |
| norm_num |
| ring |
| · simp [hn] |
| constructor |
| · simpa only [hm0,tsum_zero] using hmean.tsum_eq.symm |
| · simpa only [hc0,tsum_ite_eq,mem_univ,ite_true] using hcos.tsum_eq.symm |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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