| import Mathlib |
| import OAI.Analysis.Crouzeix.Fourier |
| import OAI.Analysis.Crouzeix.Resolvent |
| import OAI.Analysis.Crouzeix.SteinMetric |
|
|
| namespace OAI |
|
|
| /-! Disk Calculus. -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
|
|
| open scoped Topology |
|
|
| namespace CrouzeixHilbert.Disk |
|
|
| abbrev CircleSpace := AddCircle (1 : ℝ) |
|
|
| local instance : Fact (0 < (1 : ℝ)) := ⟨by norm_num⟩ |
|
|
| abbrev σ : Measure CircleSpace := AddCircle.haarAddCircle |
|
|
| section Series |
|
|
| variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] |
|
|
| omit [NormedAlgebra ℂ A] [CompleteSpace A] in |
| theorem inverse_one_sub_of_summable (a : A) (hs : Summable (fun n : ℕ => a ^ n)) : |
| IsUnit (1 - a) ∧ Ring.inverse (1 - a) = ∑' n : ℕ, a ^ n := by |
| let u : Aˣ := ⟨1 - a, ∑' n : ℕ, a ^ n, hs.one_sub_mul_tsum_pow, hs.tsum_pow_mul_one_sub⟩ |
| refine ⟨⟨u, rfl⟩, ?_⟩ |
| change Ring.inverse (u : A) = _ |
| rw [Ring.inverse_unit] |
| rfl |
|
|
| lemma norm_fourier (n : ℤ) (t : CircleSpace) : ‖fourier n t‖ = 1 := Circle.norm_coe _ |
|
|
| lemma fourier_one_pow (n : ℕ) (t : CircleSpace) : |
| (fourier 1 t) ^ n = fourier (n : ℤ) t := by |
| induction n with |
| | zero => simp |
| | succ n ih => rw [pow_succ, ih, Nat.cast_add, Nat.cast_one, fourier_add] |
|
|
| lemma inverse_fourier_pow (n : ℕ) (t : CircleSpace) : |
| ((fourier 1 t)⁻¹) ^ n = fourier (-(n : ℤ)) t := by |
| rw [Complex.inv_eq_conj (norm_fourier _ _), ← map_pow, fourier_one_pow, fourier_neg] |
|
|
| lemma summable_diskPowers (D : A) (hD : spectralRadius ℂ D < 1) (t : CircleSpace) : |
| Summable (fun n : ℕ => ((fourier 1 t)⁻¹ • D) ^ n) := by |
| apply (Metric.summable_norm_pow D hD).of_norm_bounded |
| intro n |
| simp only [smul_pow, norm_smul, norm_pow, norm_inv, norm_fourier, |
| inv_one, one_pow, one_mul, le_refl] |
|
|
| lemma disk_inverse_series (D : A) (hD : spectralRadius ℂ D < 1) (t : CircleSpace) : |
| Ring.inverse (1 - (fourier 1 t)⁻¹ • D) = |
| ∑' n : ℕ, fourier (-(n : ℤ)) t • D ^ n := by |
| rw [(inverse_one_sub_of_summable _ (summable_diskPowers D hD t)).2] |
| simp only [smul_pow, inverse_fourier_pow] |
|
|
| lemma continuous_diskInverse (D : A) (hD : spectralRadius ℂ D < 1) : |
| Continuous (fun t : CircleSpace => Ring.inverse (1 - (fourier 1 t)⁻¹ • D)) := by |
| simp_rw [disk_inverse_series D hD] |
| refine continuous_tsum (fun n => (fourier _).continuous.smul continuous_const) |
| (Metric.summable_norm_pow D hD) ?_ |
| intro n t |
| simp only [norm_smul, norm_fourier, one_mul, le_refl] |
|
|
| lemma integral_fourier (n : ℤ) : ∫ t : CircleSpace, fourier n t ∂σ = if n = 0 then 1 else 0 := by |
| have h := congrFun (@fourierCoeff_fourier 1 _ n) 0 |
| simpa only [fourierCoeff, neg_zero, fourier_zero, one_smul, Pi.single_apply, |
| eq_comm] using h |
|
|
| lemma integral_fourier_smul (a : A) (n : ℤ) : |
| ∫ t : CircleSpace, fourier n t • a ∂σ = if n = 0 then a else 0 := by |
| rw [integral_smul_const, integral_fourier] |
| split_ifs <;> simp |
|
|
| lemma integral_fourier_series (a : ℕ → A) (ha : Summable (fun n => ‖a n‖)) (k : ℕ → ℤ) : |
| (∫ t : CircleSpace, ∑' n : ℕ, fourier (k n) t • a n ∂σ) = |
| ∑' n : ℕ, if k n = 0 then a n else 0 := by |
| rw [← integral_tsum_of_summable_integral_norm] |
| · simp only [integral_fourier_smul] |
| · intro n |
| exact ((fourier _).continuous.smul continuous_const).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
| · simpa only [norm_smul, norm_fourier, one_mul, integral_const, probReal_univ, one_smul] using ha |
|
|
| end Series |
|
|
| section Density |
|
|
| variable {A : Type*} [CStarAlgebra A] |
|
|
| def phi (D : A) (t : CircleSpace) : A := |
| Ring.inverse (1 - (fourier 1 t)⁻¹ • D) + |
| star (Ring.inverse (1 - (fourier 1 t)⁻¹ • D)) - 1 |
|
|
| theorem continuous_phi (D : A) (hD : spectralRadius ℂ D < 1) : Continuous (phi D) := |
| ((continuous_diskInverse D hD).add (continuous_diskInverse D hD).star).sub continuous_const |
|
|
| theorem integrable_phi (D : A) (hD : spectralRadius ℂ D < 1) : Integrable (phi D) σ := |
| (continuous_phi D hD).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
|
|
| lemma phi_series (D : A) (hD : spectralRadius ℂ D < 1) (t : CircleSpace) : |
| phi D t = (∑' n : ℕ, fourier (-(n : ℤ)) t • D ^ n) + |
| (∑' n : ℕ, fourier (n : ℤ) t • star (D ^ n)) - 1 := by |
| simp only [phi, disk_inverse_series D hD, tsum_star, star_smul, fourier_neg, |
| starRingEnd_apply, star_star] |
|
|
| theorem integral_moment (D : A) (hD : spectralRadius ℂ D < 1) (j : ℕ) : |
| ∫ t : CircleSpace, fourier (j : ℤ) t • phi D t ∂σ = D ^ j := by |
| have hc₁ := continuous_diskInverse D hD |
| have hi₁ : Integrable (fun t : CircleSpace => Ring.inverse (1 - (fourier 1 t)⁻¹ • D)) σ := |
| hc₁.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hi₂ : Integrable (fun t : CircleSpace => |
| star (Ring.inverse (1 - (fourier 1 t)⁻¹ • D))) σ := |
| hc₁.star.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hs := Metric.summable_norm_pow D hD |
| have hs' : Summable (fun n : ℕ => ‖star (D ^ n)‖) := by simpa only [norm_star] using hs |
| have hleft : (∫ t : CircleSpace, |
| fourier (j : ℤ) t • Ring.inverse (1 - (fourier 1 t)⁻¹ • D) ∂σ) = D ^ j := by |
| simp_rw [disk_inverse_series D hD, ← tsum_const_smul'', smul_smul, ← fourier_add] |
| rw [integral_fourier_series _ hs] |
| convert tsum_ite_eq j (fun n : ℕ => D ^ n) using 1 |
| congr 1 |
| ext n |
| simp only [add_neg_eq_zero, Nat.cast_inj, eq_comm] |
| have hright : (∫ t : CircleSpace, |
| fourier (j : ℤ) t • star (Ring.inverse (1 - (fourier 1 t)⁻¹ • D)) ∂σ) = |
| if j = 0 then 1 else 0 := by |
| simp_rw [disk_inverse_series D hD, tsum_star, star_smul, fourier_neg, |
| starRingEnd_apply, star_star, ← tsum_const_smul'', smul_smul, ← fourier_add] |
| rw [integral_fourier_series _ hs'] |
| by_cases hj : j = 0 |
| · simp [hj] |
| · rw [ite_eq_right hj] |
| have hz : ∀ n : ℕ, (if (j : ℤ) + (n : ℤ) = 0 then star (D ^ n) else 0) = 0 := by |
| intro n |
| simp only [← Nat.cast_add, Nat.cast_eq_zero, Nat.add_eq_zero_iff, |
| hj, false_and, ↓reduceIte] |
| simp only [hz, tsum_zero] |
| simp only [phi, smul_sub, smul_add] |
| have hsplit := integral_sub ((hi₁.fourier_smul (j : ℤ)).add |
| (hi₂.fourier_smul (j : ℤ))) ((integrable_const (1 : A)).fourier_smul (j : ℤ)) |
| simp only [Pi.add_apply] at hsplit |
| rw [hsplit, integral_add (hi₁.fourier_smul (j : ℤ)) (hi₂.fourier_smul (j : ℤ)), |
| hleft, hright, integral_fourier_smul] |
| simp only [Nat.cast_eq_zero, add_sub_cancel_right] |
|
|
| theorem integral_phi (D : A) (hD : spectralRadius ℂ D < 1) : ∫ t, phi D t ∂σ = 1 := by |
| simpa only [Nat.cast_zero, fourier_zero, one_smul, pow_zero] using integral_moment D hD 0 |
|
|
| end Density |
|
|
| section OperatorPositivity |
|
|
| universe u |
|
|
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| theorem phi_isPositive (D : Operator H) (hn : ‖D‖ ≤ 1) (hD : spectralRadius ℂ D < 1) |
| (t : CircleSpace) : (phi D t).IsPositive := by |
| exact isPositive_diskDensity D hn (fourier 1 t) (norm_fourier _ _) |
| (inverse_one_sub_of_summable _ (summable_diskPowers D hD t)).1 |
|
|
| end OperatorPositivity |
|
|
| end CrouzeixHilbert.Disk |
|
|
| end |
|
|
| end OAI |
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|
