File size: 7,826 Bytes
ddc84c5 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 | import OAI.Analysis.DirectCrouzeix.PolynomialEnergy
namespace OAI
noncomputable section
open scoped Matrix Matrix.Norms.L2Operator Kronecker
namespace DirectCrouzeix
open scoped MatrixOrder ComplexOrder
open MeasureTheory
theorem analytic_domain_core {n m : ℕ} (hn : 0 < n) (hm : 0 < m)
(A : Matrix (Fin n) (Fin n) ℂ) (b : ℕ → Polynomial ℂ) (hb0 : b 0 = 1)
(h : BoundaryCircle → ℂ) (hh : Continuous h)
(hexpand : ∀ G : MatrixPolynomial m, ∃ (N : ℕ) (C : Fin N → Matrix (Fin m) (Fin m) ℂ),
G = faberPolynomial b C)
(hlower : ∀ N (C : Fin N → Matrix (Fin m) (Fin m) ℂ),
∑ k, hsSq (C k) ≤ polynomialEnergy h (faberPolynomial b C))
(hupper : ∀ N (C : Fin N → Matrix (Fin m) (Fin m) ℂ),
polynomialEnergy h (faberPolynomial b C) ≤ ∑ k : Fin N, faberWeight k * hsSq (C k))
(R : BoundaryCircle → Matrix (Fin n) (Fin n) ℂ) (hR : Continuous R)
(hcoeff : ∀ k : ℕ, fourierCoeff R (-(k : ℤ)) = Polynomial.aeval A (b k))
(hposfreq : ∀ k : ℤ, 0 < k → fourierCoeff R k = 0)
(hpositive : ∀ t, (R t + (R t)ᴴ).PosSemidef)
(F : MatrixPolynomial m) (hF : ∀ t, ‖matrixPolynomialValue (h t) F‖ ≤ 1) :
‖tensorPolynomial A F‖ ≤ 2 := by
let : NeZero n := ⟨Nat.ne_of_gt hn⟩
let : NeZero m := ⟨Nat.ne_of_gt hm⟩
let γ := ‖tensorPolynomial A F‖
by_cases hγ0 : γ = 0
· change γ ≤ 2
rw [hγ0]
norm_num
have hγ : 0 < γ := lt_of_le_of_ne (norm_nonneg _) (Ne.symm hγ0)
let T := Matrix.toEuclideanCLM (n := Fin n × Fin m) (𝕜 := ℂ) (tensorPolynomial A F)
have hT : ‖T‖ = γ := Matrix.l2_opNorm_toEuclideanCLM _
obtain ⟨x,y,hx,hy,hxy,hyx⟩ := exists_singular_pair T (hT ▸ hγ)
rw [hT] at hxy hyx
let X := matrixify x
let Y := matrixify y
have hFx : T (hsVector X) = (γ : ℂ) • hsVector Y := by simpa only [X,Y,hsVector_matrixify] using hxy
have hFy : ContinuousLinearMap.adjoint T (hsVector Y) = (γ : ℂ) • hsVector X := by
simpa only [X,Y,hsVector_matrixify] using hyx
let P := fun t => Xᴴ * R t * X
let Q := fun t => Yᴴ * R t * Y
let M := fun t => Yᴴ * R t * X
let L := fun t => Xᴴ * R t * Y
have hC (U V : Matrix (Fin n) (Fin m) ℂ) :
Continuous (fun t => Vᴴ * R t * U) := (compressionCLM U V).continuous.comp hR
have hCp (U V : Matrix (Fin n) (Fin m) ℂ) (k : ℤ) (hk : 0 < k) :
fourierCoeff (fun t => Vᴴ * R t * U) k = 0 := by
rw [fourierCoeff_compression U V R hR,hposfreq k hk]
simp
have hCc (U V : Matrix (Fin n) (Fin m) ℂ) (k : ℕ) :
fourierCoeff (fun t => Vᴴ * R t * U) (-(k : ℤ)) = Vᴴ * Polynomial.aeval A (b k) * U := by
rw [fourierCoeff_compression U V R hR,hcoeff]
have hC0 (U V : Matrix (Fin n) (Fin m) ℂ) :
fourierCoeff (fun t => Vᴴ * R t * U) 0 = Vᴴ * U := by
simpa only [Int.natCast_zero,neg_zero,hb0,map_one,Matrix.mul_one] using hCc U V 0
let a := ∫ t, hsSq (M t) ∂circleMeasure
have ha : 0 ≤ a := integral_nonneg (fun _ => hsSq_nonneg _)
have hapos : 0 < a := by
have hv := cross_boundary_estimate A b h hexpand hlower R hR hcoeff X Y F
change ‖inner ℂ (hsVector Y) (T (hsVector X))‖^2 ≤ a * polynomialEnergy h F at hv
have hyHS : ‖hsVector Y‖ = 1 := by simpa only [Y,hsVector_matrixify] using hy
rw [hFx,inner_smul_right,inner_self_eq_norm_sq_to_K,hyHS] at hv
by_contra hnot
have ha0 : a = 0 := le_antisymm (not_lt.mp hnot) ha
rw [ha0,zero_mul] at hv
have hv' : γ^2 ≤ 0 := by simpa using hv
nlinarith [sq_pos_of_pos hγ]
have hdiag := singular_diagonal_estimates A b h hh hexpand hlower hupper R hR hcoeff F hF X Y γ hγ hFx hFy
change (_ ≤ a/γ^2) ∧ (_ ≤ a/γ^2) at hdiag
have hPE : (∫ t, hsSq (P t + (P t)ᴴ) ∂circleMeasure) ≤ 4*a/γ^2 := by
rw [anti_diagonal_energy P (hC X X) (hCp X X)]
· have hc0 := hCc X X 0
have hcs (k : ℕ) := hCc X X (k+1)
simp only [Int.natCast_zero,neg_zero,Nat.cast_add,Nat.cast_one] at hc0 hcs
simp_rw [show P = (fun t => Xᴴ * R t * X) from rfl,hc0,hcs]
simpa only [mul_div_assoc] using mul_le_mul_of_nonneg_left hdiag.1 (by norm_num : (0 : ℝ) ≤ 4)
· rw [hC0 X X]
exact Matrix.isHermitian_conjTranspose_mul_self X
have hQE : (∫ t, hsSq (Q t + (Q t)ᴴ) ∂circleMeasure) ≤ 4*a/γ^2 := by
rw [anti_diagonal_energy Q (hC Y Y) (hCp Y Y)]
· have hc0 := hCc Y Y 0
have hcs (k : ℕ) := hCc Y Y (k+1)
simp only [Int.natCast_zero,neg_zero,Nat.cast_add,Nat.cast_one] at hc0 hcs
simp_rw [show Q = (fun t => Yᴴ * R t * Y) from rfl,hc0,hcs]
simpa only [mul_div_assoc] using mul_le_mul_of_nonneg_left hdiag.2 (by norm_num : (0 : ℝ) ≤ 4)
· rw [hC0 Y Y]
exact Matrix.isHermitian_conjTranspose_mul_self Y
have hME : a ≤ ∫ t, hsSq (M t + (L t)ᴴ) ∂circleMeasure := by
apply anti_cross_energy M L (hC X Y) (hC Y X) (hCp X Y) (hCp Y X)
rw [hC0 Y X,hC0 X Y]
simp
have hEcont (U V U' V' : Matrix (Fin n) (Fin m) ℂ) :
Continuous (fun t => hsSq (Vᴴ * R t * U + (V'ᴴ * R t * U')ᴴ)) :=
continuous_hsSq.comp ((hC U V).add (conjugateTransposeCLM.continuous.comp (hC U' V')))
have hpi := continuous_integrable_compact circleMeasure (hEcont X X X X)
have hqi := continuous_integrable_compact circleMeasure (hEcont Y Y Y Y)
have hmi := continuous_integrable_compact circleMeasure (hEcont X Y Y X)
have hblock : 2 * (∫ t, hsSq (M t + (L t)ᴴ) ∂circleMeasure) ≤
(∫ t, hsSq (P t + (P t)ᴴ) ∂circleMeasure) +
∫ t, hsSq (Q t + (Q t)ᴴ) ∂circleMeasure := by
rw [← integral_const_mul,← integral_add hpi hqi]
exact integral_mono (hmi.const_mul 2) (hpi.add hqi)
(fun t => positive_block_hsSq (positive_compression_blocks (R t) X Y (hpositive t)))
have hb : 2*a ≤ 8*a/γ^2 := by
simp only [mul_div_assoc] at hPE hQE ⊢
linarith
have hmul := (le_div_iff₀ (sq_pos_of_pos hγ)).mp hb
change γ ≤ 2
have hg2 : γ^2 ≤ 4 := by nlinarith [hapos]
nlinarith
theorem faber_expansion_of_degree {m : ℕ} (hm : 0 < m)
(b : ℕ → Polynomial ℂ) (hb : ∀ k, (b k).degree = k) (G : MatrixPolynomial m) :
∃ (N : ℕ) (C : Fin N → Matrix (Fin m) (Fin m) ℂ), G = faberPolynomial b C := by
classical
let : NeZero m := ⟨Nat.ne_of_gt hm⟩
let S : Polynomial.Sequence (Matrix (Fin m) (Fin m) ℂ) :=
⟨fun k => scalarMatrixPolynomial (b k),fun k => by
rw [scalarMatrixPolynomial,Polynomial.degree_map,hb]⟩
have hunit (k : ℕ) : IsUnit (S k).leadingCoeff := by
change IsUnit (scalarMatrixPolynomial (b k)).leadingCoeff
rw [scalarMatrixPolynomial,Polynomial.leadingCoeff_map]
apply IsUnit.map
apply isUnit_iff_ne_zero.mpr
apply Polynomial.leadingCoeff_ne_zero.mpr
apply Polynomial.degree_ne_bot.mp
rw [hb]
exact WithBot.coe_ne_bot
let N := G.natDegree+1
have himage : Set.range (fun k : Fin N => S k) = S '' Set.Iio N := by
ext p
constructor
· rintro ⟨k,rfl⟩
exact ⟨k,k.isLt,rfl⟩
· rintro ⟨k,hk,rfl⟩
exact ⟨⟨k,hk⟩,rfl⟩
have hG : G ∈ Submodule.span (Matrix (Fin m) (Fin m) ℂ) (Set.range (fun k : Fin N => S k)) := by
rw [himage,S.span_degreeLT (fun k _ => hunit k),Polynomial.mem_degreeLT]
exact lt_of_le_of_lt G.degree_le_natDegree (by exact_mod_cast Nat.lt_succ_self G.natDegree)
obtain ⟨C,hC⟩ := (Submodule.mem_span_range_iff_exists_fun (Matrix (Fin m) (Fin m) ℂ)).mp hG
refine ⟨N,C,?_⟩
rw [← hC]
simp only [faberPolynomial,Polynomial.smul_eq_C_mul]
rfl
end DirectCrouzeix
noncomputable section
open MeasureTheory Set Filter Metric
open scoped Topology Interval ENNReal NNReal ComplexConjugate
namespace DirectCrouzeix.Faber
end DirectCrouzeix.Faber
end
end
end OAI
|