| import Mathlib |
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| namespace OAI |
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| open Polynomial |
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| namespace Problem367 |
|
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| section Algebra |
| variable {A : Type*} [Ring A] |
|
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| theorem power_recurrence_of_quadratic {B Z : A} |
| (h : Z ^ 2 = 2 * B * Z - 1) (n : ℕ) : |
| Z ^ (n + 2) = 2 * B * Z ^ (n + 1) - Z ^ n := by |
| calc |
| Z ^ (n + 2) = Z ^ 2 * Z ^ n := by rw [← pow_add]; congr 1; omega |
| _ = (2 * B * Z - 1) * Z ^ n := by rw [h] |
| _ = 2 * B * Z ^ (n + 1) - Z ^ n := by |
| rw [sub_mul, one_mul, mul_assoc, ← pow_succ'] |
|
|
| variable [Algebra ℝ A] |
|
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| theorem pow_eq_chebyshev_of_quadratic {B Z : A} |
| (h : Z ^ 2 = 2 * B * Z - 1) (n : ℕ) : |
| Z ^ (n + 1) = |
| aeval B (Polynomial.Chebyshev.U ℝ (n : ℤ)) * Z - |
| aeval B (Polynomial.Chebyshev.U ℝ ((n : ℤ) - 1)) := by |
| induction n using Nat.twoStepInduction with |
| | zero => simp |
| | one => simpa [map_ofNat] using h |
| | more n ih0 ih1 => |
| rw [show n + 2 + 1 = (n + 1) + 2 by omega, |
| power_recurrence_of_quadratic h, ih1, ih0] |
| simp only [Nat.cast_add, Nat.cast_one, Nat.cast_ofNat, add_sub_cancel_right] |
| rw [Polynomial.Chebyshev.U_add_two] |
| have hind : (n : ℤ) + 2 - 1 = (n : ℤ) + 1 := by omega |
| rw [hind] |
| simp only [Polynomial.Chebyshev.U_add_one] |
| simp only [map_sub, map_mul, map_ofNat, aeval_X] |
| noncomm_ring |
|
|
| end Algebra |
|
|
| theorem abs_sin_nat_mul_le (n : ℕ) (x : ℝ) : |
| |Real.sin ((n : ℝ) * x)| ≤ (n : ℝ) * |Real.sin x| := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| rw [Nat.cast_succ, add_mul, one_mul, Real.sin_add] |
| calc |
| |Real.sin ((n : ℝ) * x) * Real.cos x + |
| Real.cos ((n : ℝ) * x) * Real.sin x| |
| ≤ |Real.sin ((n : ℝ) * x) * Real.cos x| + |
| |Real.cos ((n : ℝ) * x) * Real.sin x| := abs_add_le _ _ |
| _ = |Real.sin ((n : ℝ) * x)| * |Real.cos x| + |
| |Real.cos ((n : ℝ) * x)| * |Real.sin x| := by rw [abs_mul, abs_mul] |
| _ ≤ ((n : ℝ) * |Real.sin x|) * 1 + 1 * |Real.sin x| := by |
| apply add_le_add |
| · exact mul_le_mul ih (Real.abs_cos_le_one x) (abs_nonneg _) (by positivity) |
| · exact mul_le_mul_of_nonneg_right (Real.abs_cos_le_one _) (abs_nonneg _) |
| _ = ((n : ℝ) + 1) * |Real.sin x| := by ring |
|
|
| theorem chebyshev_U_abs_le (n : ℕ) {x : ℝ} (hx : x ∈ Set.Icc (-1) 1) : |
| |(Polynomial.Chebyshev.U ℝ (n : ℤ)).eval x| ≤ (n : ℝ) + 1 := by |
| have hn : 0 ≤ (n : ℝ) + 1 := by positivity |
| rcases eq_or_lt_of_le hx.1 with hleft | hleft |
| · subst x |
| simp [abs_mul, abs_pow, abs_of_nonneg hn] |
| rcases eq_or_lt_of_le hx.2 with hright | hright |
| · subst x |
| simp [abs_of_nonneg hn] |
| have hsin : 0 < Real.sin (Real.arccos x) := |
| Real.sin_pos_of_pos_of_lt_pi (Real.arccos_pos.mpr hright) (Real.arccos_lt_pi.mpr hleft) |
| have htrig := Polynomial.Chebyshev.U_real_cos (Real.arccos x) (n : ℤ) |
| rw [Real.cos_arccos hx.1 hx.2] at htrig |
| have hbound := abs_sin_nat_mul_le (n + 1) (Real.arccos x) |
| have habs := congrArg abs htrig |
| simp only [abs_mul, abs_of_pos hsin, Int.cast_natCast] at habs |
| apply (mul_le_mul_iff_left₀ hsin).mp |
| rw [habs] |
| simpa [Nat.cast_add, Nat.cast_one, abs_of_pos hsin] using hbound |
|
|
| theorem chebyshev_T_abs_le (n : ℕ) {x : ℝ} (hx : x ∈ Set.Icc (-1) 1) : |
| |(Polynomial.Chebyshev.T ℝ (n : ℤ)).eval x| ≤ 1 := by |
| have htrig := Polynomial.Chebyshev.T_real_cos (Real.arccos x) (n : ℤ) |
| rw [Real.cos_arccos hx.1 hx.2] at htrig |
| rw [htrig] |
| exact Real.abs_cos_le_one _ |
|
|
| section NormedRecurrence |
| variable {A : Type*} [NormedRing A] [NormedAlgebra ℝ A] |
|
|
| theorem norm_pow_le_of_chebyshev_bound {B Z : A} |
| (h : Z ^ 2 = 2 * B * Z - 1) |
| (hU : ∀ n : ℕ, ‖aeval B (Polynomial.Chebyshev.U ℝ (n : ℤ))‖ ≤ (n : ℝ) + 1) |
| (n : ℕ) : ‖Z ^ (n + 1)‖ ≤ ((n : ℝ) + 1) * ‖Z‖ + n := by |
| have hprev : ‖aeval B (Polynomial.Chebyshev.U ℝ ((n : ℤ) - 1))‖ ≤ (n : ℝ) := by |
| cases n with |
| | zero => simp |
| | succ n => simpa using hU n |
| rw [pow_eq_chebyshev_of_quadratic h] |
| calc |
| ‖aeval B (Polynomial.Chebyshev.U ℝ (n : ℤ)) * Z - |
| aeval B (Polynomial.Chebyshev.U ℝ ((n : ℤ) - 1))‖ |
| ≤ ‖aeval B (Polynomial.Chebyshev.U ℝ (n : ℤ)) * Z‖ + |
| ‖aeval B (Polynomial.Chebyshev.U ℝ ((n : ℤ) - 1))‖ := norm_sub_le _ _ |
| _ ≤ (‖aeval B (Polynomial.Chebyshev.U ℝ (n : ℤ))‖ * ‖Z‖) + n := |
| add_le_add (norm_mul_le _ _) hprev |
| _ ≤ ((n : ℝ) + 1) * ‖Z‖ + n := |
| add_le_add (mul_le_mul_of_nonneg_right (hU n) (norm_nonneg Z)) le_rfl |
|
|
| end NormedRecurrence |
|
|
| section NormedAlgebra |
| variable {A : Type*} [NormedRing A] [StarRing A] [NormedAlgebra ℝ A] |
| [CompleteSpace A] [NormOneClass A] |
| [IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint] |
|
|
| theorem norm_aeval_chebyshev_U_le (n : ℕ) {B : A} |
| (hB : IsSelfAdjoint B) (hBnorm : ‖B‖ ≤ 1) : |
| ‖aeval B (Polynomial.Chebyshev.U ℝ (n : ℤ))‖ ≤ (n : ℝ) + 1 := by |
| rw [← cfc_polynomial _ B hB] |
| apply norm_cfc_le (by positivity) |
| intro x hx |
| have hxnorm : |x| ≤ 1 := by |
| simpa only [Real.norm_eq_abs] using (spectrum.norm_le_norm_of_mem hx).trans hBnorm |
| simpa only [Real.norm_eq_abs] using chebyshev_U_abs_le n (abs_le.mp hxnorm) |
|
|
| theorem norm_aeval_chebyshev_T_le (n : ℕ) {B : A} |
| (hB : IsSelfAdjoint B) (hBnorm : ‖B‖ ≤ 1) : |
| ‖aeval B (Polynomial.Chebyshev.T ℝ (n : ℤ))‖ ≤ 1 := by |
| rw [← cfc_polynomial _ B hB] |
| apply norm_cfc_le zero_le_one |
| intro x hx |
| have hxnorm : |x| ≤ 1 := by |
| simpa only [Real.norm_eq_abs] using (spectrum.norm_le_norm_of_mem hx).trans hBnorm |
| simpa only [Real.norm_eq_abs] using chebyshev_T_abs_le n (abs_le.mp hxnorm) |
|
|
| theorem norm_pow_le_of_selfAdjoint_quadratic {B Z : A} |
| (hB : IsSelfAdjoint B) (hBnorm : ‖B‖ ≤ 1) |
| (h : Z ^ 2 = 2 * B * Z - 1) (n : ℕ) : |
| ‖Z ^ (n + 1)‖ ≤ ((n : ℝ) + 1) * ‖Z‖ + n := by |
| exact norm_pow_le_of_chebyshev_bound h |
| (fun n => norm_aeval_chebyshev_U_le n hB hBnorm) n |
|
|
| theorem norm_pow_le_linear_of_selfAdjoint_quadratic {B Z : A} |
| (hB : IsSelfAdjoint B) (hBnorm : ‖B‖ ≤ 1) |
| (h : Z ^ 2 = 2 * B * Z - 1) (n : ℕ) : |
| ‖Z ^ n‖ ≤ (‖Z‖ + 1) * ((n : ℝ) + 1) := by |
| cases n with |
| | zero => simp |
| | succ n => |
| have hn := norm_pow_le_of_selfAdjoint_quadratic hB hBnorm h n |
| simp only [Nat.cast_succ] |
| nlinarith [norm_nonneg Z] |
|
|
| theorem norm_scaled_pow_le_of_selfAdjoint_quadratic {B Z : A} {q : ℝ} |
| (hB : IsSelfAdjoint B) (hBnorm : ‖B‖ ≤ 1) |
| (h : Z ^ 2 = 2 * B * Z - 1) (hq : 0 ≤ q) (n : ℕ) : |
| ‖(q • Z) ^ n‖ ≤ (‖Z‖ + 1) * ((n : ℝ) + 1) * q ^ n := by |
| rw [_root_.smul_pow, norm_smul, Real.norm_eq_abs, abs_pow, abs_of_nonneg hq] |
| calc |
| q ^ n * ‖Z ^ n‖ ≤ q ^ n * ((‖Z‖ + 1) * ((n : ℝ) + 1)) := |
| mul_le_mul_of_nonneg_left (norm_pow_le_linear_of_selfAdjoint_quadratic hB hBnorm h n) |
| (pow_nonneg hq n) |
| _ = (‖Z‖ + 1) * ((n : ℝ) + 1) * q ^ n := by ring |
|
|
| theorem summable_norm_scaled_pow_of_selfAdjoint_quadratic {B Z : A} {q : ℝ} |
| (hB : IsSelfAdjoint B) (hBnorm : ‖B‖ ≤ 1) |
| (h : Z ^ 2 = 2 * B * Z - 1) (hq : 0 ≤ q) (hq1 : q < 1) : |
| Summable (fun n : ℕ => ‖(q • Z) ^ n‖) := by |
| have hqnorm : ‖q‖ < 1 := by simpa [Real.norm_eq_abs, abs_of_nonneg hq] using hq1 |
| have hs : Summable (fun n : ℕ => ((n : ℝ) + 1) * q ^ n) := by |
| simpa only [pow_one, add_mul, one_mul] using |
| (summable_pow_mul_geometric_of_norm_lt_one 1 hqnorm).add |
| (summable_geometric_of_norm_lt_one hqnorm) |
| apply Summable.of_nonneg_of_le (fun n => norm_nonneg _) (fun n => |
| norm_scaled_pow_le_of_selfAdjoint_quadratic hB hBnorm h hq n) |
| simpa only [mul_assoc] using hs.mul_left (‖Z‖ + 1) |
|
|
| end NormedAlgebra |
| end Problem367 |
|
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| end OAI |
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