| import OAI.Analysis.DirectCrouzeix.BoundaryKernel | |
| namespace OAI | |
| universe u_117 u_118 u_119 u_120 u_121 u_122 u_123 u_124 u_125 u_126 u_127 u_128 | |
| noncomputable section | |
| open scoped Matrix Matrix.Norms.L2Operator Kronecker | |
| noncomputable section | |
| open MeasureTheory Set Filter Metric | |
| open scoped Topology Interval ENNReal NNReal ComplexConjugate | |
| namespace DirectCrouzeix.Faber | |
| theorem weighted_integral_norm_sq_le {α : Type u_117} {E : Type u_118} [MeasurableSpace α] | |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] | |
| (μ : Measure α) (k : α → ℝ) (u : α → E) | |
| (hk : ∀ x, 0 ≤ k x) (hki : Integrable k μ) | |
| (hui : Integrable (fun x => k x • u x) μ) | |
| (hsqi : Integrable (fun x => k x * ‖u x‖ ^ 2) μ) | |
| (hmass : ∫ x, k x ∂μ = 1) : | |
| ‖∫ x, k x • u x ∂μ‖ ^ 2 ≤ ∫ x, k x * ‖u x‖ ^ 2 ∂μ := by | |
| let v := ∫ x, k x • u x ∂μ | |
| have hii : Integrable (fun x => inner ℝ v (k x • u x)) μ := | |
| (innerSL ℝ v).integrable_comp hui | |
| have hi : (∫ x, inner ℝ v (k x • u x) ∂μ) = ‖v‖ ^ 2 := by | |
| change (∫ x, (innerSL ℝ v) (k x • u x) ∂μ) = _ | |
| rw [(innerSL ℝ v).integral_comp_comm hui] | |
| exact real_inner_self_eq_norm_sq v | |
| have hn : 0 ≤ ∫ x, k x * ‖u x - v‖ ^ 2 ∂μ := | |
| integral_nonneg (fun x => mul_nonneg (hk x) (sq_nonneg _)) | |
| have hexp : (fun x => k x * ‖u x - v‖ ^ 2) = | |
| fun x => (k x * ‖u x‖ ^ 2 - 2 * inner ℝ v (k x • u x)) + k x * ‖v‖ ^ 2 := by | |
| funext x | |
| rw [norm_sub_sq_real, inner_smul_right, real_inner_comm v (u x)] | |
| ring | |
| rw [hexp] at hn | |
| have hsubi : Integrable (fun x => k x * ‖u x‖ ^ 2 - 2 * inner ℝ v (k x • u x)) μ := | |
| hsqi.sub (hii.const_mul 2) | |
| have hconsti : Integrable (fun x => k x * ‖v‖ ^ 2) μ := hki.mul_const _ | |
| rw [integral_add hsubi hconsti, integral_sub hsqi (hii.const_mul 2), | |
| integral_const_mul, integral_mul_const, hi, hmass] at hn | |
| change ‖v‖ ^ 2 ≤ _ | |
| linarith | |
| theorem continuous_integrable_compact {α : Type u_119} {E : Type u_120} [TopologicalSpace α] | |
| [MeasurableSpace α] [OpensMeasurableSpace α] [CompactSpace α] | |
| [NormedAddCommGroup E] (μ : Measure α) [IsFiniteMeasure μ] | |
| {f : α → E} (hf : Continuous f) : Integrable f μ := | |
| hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f) | |
| theorem continuous_integral_compact {α : Type u_121} {β : Type u_122} {E : Type u_123} | |
| [TopologicalSpace α] [FirstCountableTopology α] [LocallyCompactSpace α] | |
| [TopologicalSpace β] [MeasurableSpace β] [OpensMeasurableSpace β] | |
| [CompactSpace β] [SecondCountableTopology β] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| (μ : Measure β) [IsFiniteMeasure μ] {f : α → β → E} | |
| (hf : Continuous f.uncurry) : Continuous (fun x => ∫ y, f x y ∂μ) := by | |
| simpa using continuous_parametric_integral_of_continuous (μ := μ) hf isCompact_univ | |
| theorem markov_kernel_contraction {α : Type u_124} {β : Type u_125} {E : Type u_126} | |
| [MetricSpace α] [MeasurableSpace α] [BorelSpace α] [CompactSpace α] | |
| [SecondCountableTopology α] | |
| [MetricSpace β] [MeasurableSpace β] [BorelSpace β] [CompactSpace β] | |
| [SecondCountableTopology β] | |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] | |
| (μ : Measure α) (ν : Measure β) [IsFiniteMeasure μ] [IsFiniteMeasure ν] | |
| (K : α → β → ℝ) (u : α → E) | |
| (hK : Continuous K.uncurry) (hu : Continuous u) | |
| (hpos : ∀ x y, 0 ≤ K x y) | |
| (hcol : ∀ y, ∫ x, K x y ∂μ = 1) | |
| (hrow : ∀ x, ∫ y, K x y ∂ν = 1) : | |
| (∫ y, ‖∫ x, K x y • u x ∂μ‖ ^ 2 ∂ν) ≤ ∫ x, ‖u x‖ ^ 2 ∂μ := by | |
| have hKcol (y : β) : Continuous (fun x => K x y) := | |
| hK.comp (continuous_id.prodMk continuous_const) | |
| have hT : Continuous (fun y => ∫ x, K x y • u x ∂μ) := by | |
| apply continuous_integral_compact | |
| exact (hK.comp continuous_swap).smul (hu.comp continuous_snd) | |
| have hs : Integrable (fun p : α × β => K p.1 p.2 * ‖u p.1‖ ^ 2) (μ.prod ν) := | |
| continuous_integrable_compact _ (hK.mul ((hu.comp continuous_fst).norm.pow 2)) | |
| calc | |
| _ ≤ ∫ y, ∫ x, K x y * ‖u x‖ ^ 2 ∂μ ∂ν := by | |
| apply integral_mono (continuous_integrable_compact _ (hT.norm.pow 2)) | |
| hs.integral_prod_right | |
| intro y | |
| exact weighted_integral_norm_sq_le μ (fun x => K x y) u (fun x => hpos x y) | |
| (continuous_integrable_compact _ (hKcol y)) | |
| (continuous_integrable_compact _ ((hKcol y).smul hu)) | |
| (continuous_integrable_compact _ ((hKcol y).mul (hu.norm.pow 2))) (hcol y) | |
| _ = ∫ x, ∫ y, K x y * ‖u x‖ ^ 2 ∂ν ∂μ := (integral_integral_swap hs).symm | |
| _ = _ := by simp_rw [integral_mul_const, hrow, one_mul] | |
| section Hilbert | |
| variable {E : Type u_127} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] | |
| def positiveSeries {N : ℕ} (C : Fin N → E) (t : Angle) : E := | |
| ∑ k : Fin N, fourier (k:ℤ) t • C k | |
| theorem continuous_positiveSeries {E : Type u_127} [NormedAddCommGroup E] | |
| [InnerProductSpace ℂ E] [CompleteSpace E] {N : ℕ} (C : Fin N → E) : | |
| Continuous (positiveSeries C) := by | |
| exact continuous_finsetSum _ (fun k _ => (fourier (k:ℤ)).continuous.smul continuous_const) | |
| theorem fourierCoeff_finite_smul {N : ℕ} (f : Fin N → Angle → ℂ) | |
| (hf : ∀ k, Continuous (f k)) (C : Fin N → E) (j : ℤ) : | |
| fourierCoeff (fun t => ∑ k, f k t • C k) j = ∑ k, fourierCoeff (f k) j • C k := by | |
| unfold fourierCoeff | |
| simp_rw [Finset.smul_sum,smul_smul] | |
| rw [integral_finsetSum] | |
| · simp_rw [integral_smul_const,smul_eq_mul] | |
| · intro k hk | |
| exact integrable_of_continuous_compact _ (((fourier (-j)).continuous.mul (hf k)).smul continuous_const) | |
| theorem fourierCoeff_positiveSeries {N : ℕ} (C : Fin N → E) (j : Fin N) : | |
| fourierCoeff (positiveSeries C) (j:ℤ) = C j := by | |
| classical | |
| unfold positiveSeries | |
| rw [fourierCoeff_finite_smul _ (fun k => (fourier (k:ℤ)).continuous)] | |
| simp_rw [fourierCoeff_fourier] | |
| simp only [Pi.single_apply,ite_smul,one_smul,zero_smul] | |
| have hinj (k : Fin N) : (j:ℤ) = (k:ℤ) ↔ j = k := by | |
| constructor | |
| · intro h; apply Fin.ext; exact_mod_cast h | |
| · intro h; rw [h] | |
| simp_rw [hinj] | |
| simp | |
| theorem integral_inner_positiveSeries {N : ℕ} (C : Fin N → E) | |
| (f : Angle → E) (hf : Continuous f) : | |
| (∫ t, inner ℂ (positiveSeries C t) (f t) ∂angularMeasure) = | |
| ∑ k, inner ℂ (C k) (fourierCoeff f (k:ℤ)) := by | |
| simp only [positiveSeries,sum_inner,inner_smul_left,← fourier_neg] | |
| rw [integral_finsetSum] | |
| · apply Finset.sum_congr rfl | |
| intro k hk | |
| have he : (fun t => fourier (-(k:ℤ)) t * inner ℂ (C k) (f t)) = | |
| (fun t => inner ℂ (C k) (fourier (-(k:ℤ)) t • f t)) := by | |
| funext t; simp only [inner_smul_right] | |
| rw [he,integral_inner] | |
| · rfl | |
| · exact integrable_of_continuous_compact _ ((fourier (-(k:ℤ))).continuous.smul hf) | |
| · intro k hk | |
| exact integrable_of_continuous_compact _ | |
| ((fourier (-(k:ℤ))).continuous.mul (continuous_const.inner hf)) | |
| theorem energy_positiveSeries {N : ℕ} (C : Fin N → E) : | |
| (∫ t, ‖positiveSeries C t‖^2 ∂angularMeasure) = ∑ k, ‖C k‖^2 := by | |
| have hp := continuous_positiveSeries C | |
| have he (t : Angle) : ‖positiveSeries C t‖^2 = RCLike.re (inner ℂ (positiveSeries C t) (positiveSeries C t)) := by | |
| simp [inner_self_eq_norm_sq_to_K,← Complex.ofReal_pow] | |
| simp_rw [he] | |
| rw [integral_re (integrable_of_continuous_compact _ (hp.inner hp)),integral_inner_positiveSeries C _ hp] | |
| simp only [fourierCoeff_positiveSeries,inner_self_eq_norm_sq_to_K,map_sum] | |
| change (∑ k, ((‖C k‖ : ℂ)^2).re) = ∑ k, ‖C k‖^2 | |
| simp [← Complex.ofReal_pow] | |
| theorem energy_orthogonal_positiveSeries {N : ℕ} (C : Fin N → E) | |
| (r : Angle → E) (hr : Continuous r) (hfreq : ∀ j : ℤ, 0 ≤ j → fourierCoeff r j = 0) : | |
| (∫ t, ‖positiveSeries C t+r t‖^2 ∂angularMeasure) = | |
| (∑ k, ‖C k‖^2) + ∫ t, ‖r t‖^2 ∂angularMeasure := by | |
| have hp := continuous_positiveSeries C | |
| have hci : Integrable (fun t => inner ℂ (positiveSeries C t) (r t)) angularMeasure := | |
| integrable_of_continuous_compact _ (hp.inner hr) | |
| have hcross : (∫ t, inner ℂ (positiveSeries C t) (r t) ∂angularMeasure) = 0 := by | |
| rw [integral_inner_positiveSeries C r hr] | |
| apply Finset.sum_eq_zero | |
| intro k hk | |
| rw [hfreq (k:ℤ) (by positivity),inner_zero_right] | |
| have hcrossRe : (∫ t, RCLike.re (inner ℂ (positiveSeries C t) (r t)) ∂angularMeasure) = 0 := by | |
| rw [integral_re hci,hcross] | |
| rfl | |
| have hpi : Integrable (fun t => ‖positiveSeries C t‖^2) angularMeasure := | |
| integrable_of_continuous_compact _ (hp.norm.pow 2) | |
| have hri : Integrable (fun t => ‖r t‖^2) angularMeasure := integrable_of_continuous_compact _ (hr.norm.pow 2) | |
| have hcii : Integrable (fun t => 2*RCLike.re (inner ℂ (positiveSeries C t) (r t))) angularMeasure := | |
| hci.re.const_mul 2 | |
| simp_rw [norm_add_sq (𝕜 := ℂ)] | |
| have ha := integral_add (hpi.add hcii) hri | |
| have hb := integral_add hpi hcii | |
| simp only [Pi.add_apply] at ha hb | |
| rw [ha,hb,integral_const_mul,hcrossRe,mul_zero,add_zero,energy_positiveSeries] | |
| end Hilbert | |
| section FaberHilbert | |
| variable {E : Type u_128} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] | |
| end FaberHilbert | |
| end DirectCrouzeix.Faber | |
| end | |
| end | |
| end OAI | |