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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 | import Mathlib
namespace OAI
open Polynomial
namespace Problem367
section Algebra
variable {A : Type*} [Ring A]
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]
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
end OAI
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