| import Mathlib |
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| namespace OAI |
|
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| open Polynomial |
| namespace Problem367 |
|
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| 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] |
|
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| 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 _) |
|
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| 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] |
|
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| 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 |
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| end OAI |
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