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b317718 | 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 | import Mathlib
namespace OAI
open Polynomial
namespace Problem367
lemma chebyshev_pell (n : ℕ) :
Chebyshev.T ℝ ((n : ℤ) + 1) ^ 2 +
(1 - (X : ℝ[X]) ^ 2) * Chebyshev.U ℝ (n : ℤ) ^ 2 = 1 := by
induction n with
| zero => simp
| succ n ih =>
simp only [Nat.cast_succ]
rw [show (n : ℤ) + 1 + 1 = (n : ℤ) + 2 by ring,
Chebyshev.T_eq_X_mul_T_sub_pol_U, Chebyshev.U_eq_X_mul_U_add_T]
calc
_ = Chebyshev.T ℝ ((n : ℤ) + 1) ^ 2 +
(1 - (X : ℝ[X]) ^ 2) * Chebyshev.U ℝ (n : ℤ) ^ 2 := by ring
_ = 1 := ih
end Problem367
namespace Problem367
section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
lemma isSelfAdjoint_aeval_real (B : E →L[ℝ] E) (hB : IsSelfAdjoint B) (p : ℝ[X]) :
IsSelfAdjoint (aeval B p) := by
induction p using Polynomial.induction_on with
| C r =>
simpa using (show IsSelfAdjoint r from rfl).algebraMap (E →L[ℝ] E)
| add p q hp hq => simpa using hp.add hq
| monomial n r ih =>
simp only [map_mul, map_pow, aeval_C, aeval_X]
rw [← Algebra.smul_def]
exact (show IsSelfAdjoint r from rfl).smul (hB.pow _)
omit [CompleteSpace E] in
lemma commute_aeval_self (B : E →L[ℝ] E) (p : ℝ[X]) : Commute B (aeval B p) := by
simpa using (Commute.all (X : ℝ[X]) p).map (aeval B)
end
end Problem367
namespace Problem367
section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
lemma selfAdjoint_inner_square (P : E →L[ℝ] E) (hP : IsSelfAdjoint P) (x : E) :
inner ℝ x ((P ^ 2) x) = ‖P x‖ ^ 2 := by
rw [pow_two, mul_apply_eq_comp]
exact ((hP.isSymmetric x (P x)).symm).trans (real_inner_self_eq_norm_sq _)
lemma norm_le_one_of_pell (B P Q : E →L[ℝ] E)
(hB : IsSelfAdjoint B) (hP : IsSelfAdjoint P) (hQ : IsSelfAdjoint Q)
(hBQ : Commute B Q) (hBnorm : ‖B‖ ≤ 1)
(hidentity : P ^ 2 + (1 - B ^ 2) * Q ^ 2 = 1) : ‖P‖ ≤ 1 := by
have hBQsa : IsSelfAdjoint (B * Q) := (hB.commute_iff hQ).mp hBQ
have hid : P ^ 2 + Q ^ 2 - (B * Q) ^ 2 = 1 := by
rw [hBQ.mul_pow]
calc
P ^ 2 + Q ^ 2 - B ^ 2 * Q ^ 2 = P ^ 2 + (1 - B ^ 2) * Q ^ 2 := by noncomm_ring
_ = 1 := hidentity
refine P.opNorm_le_bound (by norm_num) fun x => ?_
have heq := congrArg (fun A : E →L[ℝ] E => inner ℝ x (A x)) hid
simp only [sub_apply, add_apply,
one_apply_eq_self, inner_sub_right, inner_add_right,
selfAdjoint_inner_square P hP x, selfAdjoint_inner_square Q hQ x,
selfAdjoint_inner_square (B * Q) hBQsa x, real_inner_self_eq_norm_sq,
mul_apply_eq_comp] at heq
have hb : ‖B (Q x)‖ ≤ ‖Q x‖ := by
calc
‖B (Q x)‖ ≤ ‖B‖ * ‖Q x‖ := B.le_opNorm _
_ ≤ 1 * ‖Q x‖ := mul_le_mul_of_nonneg_right hBnorm (norm_nonneg _)
_ = ‖Q x‖ := one_mul _
have hb2 : ‖B (Q x)‖ ^ 2 ≤ ‖Q x‖ ^ 2 :=
pow_le_pow_left₀ (norm_nonneg _) hb 2
simp only [one_mul]
apply (sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).mp
linarith
end
end Problem367
namespace Problem367
section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
lemma norm_aeval_chebyshev_T_le_real (B : E →L[ℝ] E)
(hB : IsSelfAdjoint B) (hBnorm : ‖B‖ ≤ 1) (n : ℕ) :
‖aeval B (Chebyshev.T ℝ (n : ℤ))‖ ≤ 1 := by
cases n with
| zero =>
simp only [Nat.cast_zero, Chebyshev.T_zero, map_one]
exact ContinuousLinearMap.norm_id_le
| succ n =>
simp only [Nat.cast_succ]
apply norm_le_one_of_pell B _ (aeval B (Chebyshev.U ℝ (n : ℤ))) hB
(isSelfAdjoint_aeval_real B hB _) (isSelfAdjoint_aeval_real B hB _)
(commute_aeval_self B _) hBnorm
have h := congrArg (aeval B) (chebyshev_pell n)
simpa only [map_add, map_mul, map_sub, map_pow, map_one, aeval_X] using h
lemma norm_aeval_chebyshev_U_le_real (B : E →L[ℝ] E)
(hB : IsSelfAdjoint B) (hBnorm : ‖B‖ ≤ 1) (n : ℕ) :
‖aeval B (Chebyshev.U ℝ (n : ℤ))‖ ≤ (n : ℝ) + 1 := by
induction n using Nat.twoStepInduction with
| zero =>
simp only [Nat.cast_zero, Chebyshev.U_zero, map_one, zero_add]
exact ContinuousLinearMap.norm_id_le
| one =>
simpa [Chebyshev.U_one, map_ofNat, two_mul] using
(norm_add_le B B).trans (add_le_add hBnorm hBnorm)
| more n ih0 ih1 =>
simp only [Nat.cast_add, Nat.cast_ofNat, Chebyshev.U_eq_two_mul_T_add_U,
map_add, two_mul]
have hT := norm_aeval_chebyshev_T_le_real B hB hBnorm (n + 2)
simp only [Nat.cast_add, Nat.cast_ofNat] at hT
calc
_ ≤ (‖aeval B (Chebyshev.T ℝ ((n : ℤ) + 2))‖ +
‖aeval B (Chebyshev.T ℝ ((n : ℤ) + 2))‖) +
‖aeval B (Chebyshev.U ℝ (n : ℤ))‖ :=
(norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl)
_ ≤ (n : ℝ) + 2 + 1 := by linarith
end
end Problem367
end OAI
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