| import OAI.Analysis.DirectCrouzeix.FaberEnergy |
| |
| namespace OAI |
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| universe u_129 u_130 |
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| noncomputable section |
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| open scoped Matrix Matrix.Norms.L2Operator Kronecker |
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| noncomputable section |
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| open MeasureTheory Set Filter Metric |
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| open scoped Topology Interval ENNReal NNReal ComplexConjugate |
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| namespace DirectCrouzeix.Faber |
| |
| section BanachAlgebra |
| |
| variable {𝔄 : Type u_129} [NormedRing 𝔄] [NormedAlgebra ℂ 𝔄] [CompleteSpace 𝔄] |
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| theorem analyticAt_ringInverse {x : 𝔄} (hx : IsUnit x) : |
| AnalyticAt ℂ Ring.inverse x := by |
| rcases hx with ⟨u,rfl⟩ |
| exact analyticAt_inverse u |
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| theorem ringInverse_smul {𝔄 : Type u_129} [NormedRing 𝔄] [NormedAlgebra ℂ 𝔄] |
| [CompleteSpace 𝔄] {c : ℂ} (hc : c ≠ 0) (x : 𝔄) : |
| Ring.inverse (c • x) = c⁻¹ • Ring.inverse x := by |
| have hu : IsUnit (algebraMap ℂ 𝔄 c) := (isUnit_iff_ne_zero.mpr hc).map (algebraMap ℂ 𝔄) |
| have hi : Ring.inverse (algebraMap ℂ 𝔄 c) = algebraMap ℂ 𝔄 c⁻¹ := by |
| have hm : algebraMap ℂ 𝔄 c * algebraMap ℂ 𝔄 c⁻¹ = 1 := by |
| rw [← map_mul,mul_inv_cancel₀ hc,map_one] |
| calc |
| Ring.inverse (algebraMap ℂ 𝔄 c) = Ring.inverse (algebraMap ℂ 𝔄 c) * |
| (algebraMap ℂ 𝔄 c * algebraMap ℂ 𝔄 c⁻¹) := by rw [hm,mul_one] |
| _ = algebraMap ℂ 𝔄 c⁻¹ := by rw [← mul_assoc,Ring.inverse_mul_cancel _ hu,one_mul] |
| rw [Algebra.smul_def,Ring.inverse_mul (Or.inl hu),hi,Algebra.smul_def,Algebra.commutes] |
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| theorem ringInverse_geometric {𝔄 : Type u_129} [NormedRing 𝔄] [NormedAlgebra ℂ 𝔄] |
| [CompleteSpace 𝔄] (x : 𝔄) (hx : IsUnit (1-x)) (k : ℕ) : |
| Ring.inverse (1-x) = (∑ j ∈ Finset.range (k+1),x^j) + x^(k+1)*Ring.inverse (1-x) := by |
| have he := congrArg (fun y : 𝔄 => y * Ring.inverse (1-x)) (geom_sum_mul_neg x (k+1)) |
| rw [mul_assoc,Ring.mul_inverse_cancel _ hx,mul_one,sub_mul,one_mul] at he |
| exact (eq_add_of_sub_eq he.symm) |
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|
| theorem taylorCoeff_pow_smul_zero {E : Type u_130} [NormedAddCommGroup E] [NormedSpace ℂ E] |
| [CompleteSpace E] {f : ℂ → E} (hf : AnalyticAt ℂ f 0) {j k : ℕ} (hjk : k < j) : |
| taylorCoeff (fun u => u^j • f u) k = 0 := by |
| have he := iteratedDerivWithin_smul (𝕜 := ℂ) (𝔸 := ℂ) (F := E) (n := k) |
| (x := 0) (s := univ) (f := fun u : ℂ => u^j) (g := f) |
| (mem_univ _) uniqueDiffOn_univ (by fun_prop) hf.contDiffAt.contDiffWithinAt |
| simp only [iteratedDerivWithin_univ] at he |
| change (k.factorial : ℂ)⁻¹ • iteratedDeriv k ((fun u : ℂ => u^j) • f) 0 = 0 |
| rw [he] |
| have hz : (∑ i ∈ Finset.range (k+1), k.choose i • |
| iteratedDeriv i (fun u : ℂ => u^j) 0 • iteratedDeriv (k-i) f 0) = 0 := by |
| apply Finset.sum_eq_zero |
| intro i hi |
| have hij : i ≠ j := by have := Finset.mem_range.mp hi |
| simp only [iteratedDeriv_fun_pow_zero,ite_eq_right hij,Nat.cast_zero,smul_zero,zero_smul] |
| rw [hz,smul_zero] |
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| def algebraDenominator (c : ℂ) (g : ℂ → ℂ) (A : 𝔄) (u : ℂ) : 𝔄 := |
| (c+u*g u) • 1-u • A |
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| def algebraGenerating (c : ℂ) (g : ℂ → ℂ) (A : 𝔄) (u : ℂ) : 𝔄 := |
| (c-u^2*deriv g u) • Ring.inverse (algebraDenominator c g A u) |
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| theorem analyticAt_algebraDenominator {𝔄 : Type u_129} [NormedRing 𝔄] |
| [NormedAlgebra ℂ 𝔄] [CompleteSpace 𝔄] {c : ℂ} {g : ℂ → ℂ} {u : ℂ} |
| (hg : AnalyticAt ℂ g u) (A : 𝔄) : AnalyticAt ℂ (algebraDenominator c g A) u := by |
| exact ((analyticAt_const.add (analyticAt_id.mul hg)).smul analyticAt_const).sub |
| (analyticAt_id.smul analyticAt_const) |
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| theorem analyticAt_algebraGenerating {c : ℂ} {g : ℂ → ℂ} {u : ℂ} |
| (hg : AnalyticAt ℂ g u) (A : 𝔄) (hu : IsUnit (algebraDenominator c g A u)) : |
| AnalyticAt ℂ (algebraGenerating c g A) u := |
| (analyticAt_const.sub ((analyticAt_id.pow 2).mul hg.deriv)).smul |
| ((analyticAt_ringInverse hu).comp (analyticAt_algebraDenominator hg A)) |
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| theorem algebraGenerating_local {c : ℂ} {g : ℂ → ℂ} (A : 𝔄) {u : ℂ} |
| (h : c+u*g u ≠ 0) : |
| algebraGenerating c g A u = localN c g u • Ring.inverse (1-localW c g u • A) := by |
| have he : algebraDenominator c g A u = (c+u*g u) • (1-localW c g u • A) := by |
| simp only [algebraDenominator,smul_sub,smul_smul,localW,localL] |
| congr 1 |
| congr 1 |
| field_simp |
| rw [algebraGenerating,he,ringInverse_smul h,smul_smul] |
| rfl |
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| theorem algebraGenerating_coefficient {c : ℂ} (hc : c ≠ 0) {g : ℂ → ℂ} |
| (hg : AnalyticAt ℂ g 0) (A : 𝔄) (k : ℕ) : |
| taylorCoeff (algebraGenerating c g A) k = Polynomial.aeval A (faberBasis c g k) := by |
| classical |
| let L := localL c g |
| let W := localW c g |
| let N := localN c g |
| let V := fun u => Ring.inverse (1-W u • A) |
| have hL : AnalyticAt ℂ L 0 := analyticAt_localL hc hg |
| have hW : AnalyticAt ℂ W 0 := analyticAt_localW hc hg |
| have hN : AnalyticAt ℂ N 0 := analyticAt_localN hc hg |
| have hd : AnalyticAt ℂ (fun u => (1:𝔄)-W u • A) 0 := |
| analyticAt_const.sub (hW.smul analyticAt_const) |
| have hunit : IsUnit ((1:𝔄)-W 0 • A) := by simp [W] |
| have hV : AnalyticAt ℂ V 0 := (analyticAt_ringInverse hunit).comp (f := fun u => (1:𝔄)-W u • A) hd |
| have hlocal : ∀ᶠ u in 𝓝 0, c+u*g u ≠ 0 := |
| (analyticAt_const.add (analyticAt_id.mul hg)).continuousAt.eventually_ne (by simpa using hc) |
| have hu : ∀ᶠ u in 𝓝 0, IsUnit ((1:𝔄)-W u • A) := |
| hd.continuousAt.preimage_mem_nhds (Units.isOpen.mem_nhds hunit) |
| let T := fun u => (N u*L u^(k+1)) • (A^(k+1)*V u) |
| have hT : AnalyticAt ℂ T 0 := (hN.mul (hL.pow (k+1))).smul (analyticAt_const.mul hV) |
| have hs : AnalyticAt ℂ (fun u => ∑ j ∈ Finset.range (k+1), (N u*W u^j) • A^j) 0 := |
| Finset.analyticAt_fun_sum _ (fun j _ => (hN.mul (hW.pow j)).smul analyticAt_const) |
| have he : algebraGenerating c g A =ᶠ[𝓝 0] |
| (fun u => (∑ j ∈ Finset.range (k+1), (N u*W u^j) • A^j) + u^(k+1) • T u) := by |
| filter_upwards [hlocal,hu] with u hlu huu |
| rw [algebraGenerating_local A hlu,ringInverse_geometric (W u • A) huu k] |
| simp only [smul_add,Finset.smul_sum,smul_pow,smul_smul,smul_mul_assoc] |
| congr 1 |
| change (N u * W u^(k+1)) • (A^(k+1)*V u) = |
| u^(k+1) • ((N u*L u^(k+1)) • (A^(k+1)*V u)) |
| rw [smul_smul] |
| congr 1 |
| change N u * (u*L u)^(k+1) = _ |
| rw [mul_pow] |
| ring |
| have htail : AnalyticAt ℂ (fun u => u^(k+1) • T u) 0 := (analyticAt_id.pow (k+1)).smul hT |
| rw [taylorCoeff_congr he k,taylorCoeff_add hs htail, |
| taylorCoeff_pow_smul_zero hT (by omega : k < k+1),add_zero,taylorCoeff_sum] |
| · simp only [faberBasis,map_sum,Polynomial.aeval_monomial,← Algebra.smul_def] |
| apply Finset.sum_congr rfl |
| intro j hj |
| exact taylorCoeff_smul_const (hN.mul (hW.pow j)) (A^j) k |
| · exact fun j _ => (hN.mul (hW.pow j)).smul analyticAt_const |
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| theorem algebraGenerating_fourier {R : ℝ} (hR : 1 < R) {c : ℂ} (hc : c ≠ 0) |
| {g : ℂ → ℂ} (hg : AnalyticOnNhd ℂ g (ball 0 R)) (A : 𝔄) |
| (hunit : ∀ u ∈ closedBall 0 1, IsUnit (algebraDenominator c g A u)) : |
| (Continuous (fun t : Angle => algebraGenerating c g A (unitPoint t)⁻¹)) ∧ |
| (∀ k : ℕ, fourierCoeff (fun t : Angle => algebraGenerating c g A (unitPoint t)⁻¹) |
| (-(k:ℤ)) = Polynomial.aeval A (faberBasis c g k)) ∧ |
| (∀ k : ℤ, 0 < k → fourierCoeff (fun t : Angle => algebraGenerating c g A (unitPoint t)⁻¹) k = 0) := by |
| have ha : AnalyticOnNhd ℂ (algebraGenerating c g A) (closedBall 0 1) := by |
| intro u hu |
| exact analyticAt_algebraGenerating (hg u (closedBall_subset_ball hR hu)) A (hunit u hu) |
| obtain ⟨δ,hδ,hsub⟩ := (isCompact_closedBall (0:ℂ) 1).exists_thickening_subset_open |
| (isOpen_analyticAt ℂ (algebraGenerating c g A)) ha |
| rw [thickening_closedBall hδ (by norm_num : (0:ℝ) ≤ 1)] at hsub |
| have hδ1 : 1 < δ+1 := by linarith |
| have hsub : AnalyticOnNhd ℂ (algebraGenerating c g A) (ball 0 (δ+1)) := hsub |
| have he (k : ℤ) := fourierCoeff_of_antianalytic hδ1 hsub k |
| refine ⟨?_,?_,?_⟩ |
| · exact hsub.continuousOn.comp_continuous (unitPoint_continuous.inv₀ unitPoint_ne_zero) |
| (fun t => unitPoint_inv_mem hδ1 t) |
| · intro k |
| rw [he,ite_eq_left (by omega : -(k:ℤ) ≤ 0)] |
| simp only [neg_neg,Int.toNat_natCast] |
| exact algebraGenerating_coefficient hc (hg 0 (mem_ball_self (by linarith))) A k |
| · intro k hk |
| rw [he,ite_eq_right (by omega : ¬k ≤ 0)] |
|
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| theorem algebraDenominator_exterior {𝔄 : Type u_129} [NormedRing 𝔄] |
| [NormedAlgebra ℂ 𝔄] [CompleteSpace 𝔄] |
| (c : ℂ) (g : ℂ → ℂ) (A : 𝔄) {u : ℂ} (hu : u ≠ 0) : |
| algebraDenominator c g A u = u • (exterior c g u⁻¹ • 1 - A) := by |
| simp only [algebraDenominator,exterior,inv_inv,smul_sub,smul_smul] |
| congr 1 |
| congr 1 |
| field_simp |
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| theorem algebraGenerating_exterior (c : ℂ) (g : ℂ → ℂ) (A : 𝔄) {ζ : ℂ} (hζ : ζ ≠ 0) : |
| algebraGenerating c g A ζ⁻¹ = (ζ*(c-ζ⁻¹^2*deriv g ζ⁻¹)) • |
| Ring.inverse (exterior c g ζ • 1 - A) := by |
| rw [algebraGenerating,algebraDenominator_exterior c g A (inv_ne_zero hζ),inv_inv, |
| ringInverse_smul (inv_ne_zero hζ),inv_inv,smul_smul,mul_comm] |
| |
| end BanachAlgebra |
| |
| end DirectCrouzeix.Faber |
| |
| namespace DirectCrouzeix |
| |
| open MeasureTheory Set Filter Metric |
|
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| open scoped Topology ComplexConjugate MatrixOrder ComplexOrder |
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| end DirectCrouzeix |
| |
| end |
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| end |
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| end OAI |
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