File size: 3,998 Bytes
d5b562f | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingApproximationPower
namespace OAI
/-! # Compact approximation of the actual two scalar linearization coefficients -/
open Filter Topology
namespace DefocusingNLS
variable (a k M : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k)
(L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (hLinf : Tendsto L atTop atTop)
(q : ℕ → FourierL2) (hq : ExpandingCompactApproximation a k ha1 hk L hL q)
(hqb : ∀ n, ‖q n‖ ≤ M)
include hLinf hq hqb
theorem ExpandingCompactApproximation.circular (m : ℕ) :
ExpandingCompactApproximation a k ha1 hk L hL
(fun n => expandingCircularCoefficient a k (L n) ha ha1 hk (hL n) (m + 1) (q n)) := by
have hqp := ExpandingCompactApproximation.positivePower a k M ha ha1 hk L hL hLinf q hq hqb m
have hqc := ExpandingCompactApproximation.conjugate a k ha ha1 hk L hL q hq
have hqcb (n : ℕ) : ‖fourierConjugate (q n)‖ ≤ M := by simpa only [fourierConjugate_norm] using hqb n
have hqcp := ExpandingCompactApproximation.positivePower a k M ha ha1 hk L hL hLinf
(fun n => fourierConjugate (q n)) hqc hqcb m
let B := expandingAlgebraBound a k ^ m * M ^ (m + 1)
have hb (n : ℕ) (u : FourierL2) (hu : ‖u‖ ≤ M) :
‖expandingPower a k (L n) ha ha1 hk (hL n) u (m + 1)‖ ≤ B :=
(expandingPower_positive_norm_le a k (L n) ha ha1 hk (hL n) u m).trans
(mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) hu _)
(pow_nonneg (expandingAlgebraBound_nonneg a k) _))
have hp := ExpandingCompactApproximation.mul a k B ha ha1 hk L hL hLinf _ _ hqp hqcp
(fun n => hb n (q n) (hqb n)) (fun n => hb n (fourierConjugate (q n)) (hqcb n))
exact ExpandingCompactApproximation.smul a k ha1 hk L hL _ hp (((m + 1 + 1 : ℕ) : ℂ))
theorem ExpandingCompactApproximation.anticircular_one :
ExpandingCompactApproximation a k ha1 hk L hL
(fun n => expandingAnticircularCoefficient a k (L n) ha ha1 hk (hL n) 1 (q n)) := by
simpa only [expandingAnticircularCoefficient_one] using
ExpandingCompactApproximation.positivePower a k M ha ha1 hk L hL hLinf q hq hqb 1
theorem ExpandingCompactApproximation.anticircular_succ (m : ℕ) :
ExpandingCompactApproximation a k ha1 hk L hL
(fun n => expandingAnticircularCoefficient a k (L n) ha ha1 hk (hL n) (m + 2) (q n)) := by
have hqp := ExpandingCompactApproximation.positivePower a k M ha ha1 hk L hL hLinf q hq hqb (m + 2)
have hqc := ExpandingCompactApproximation.conjugate a k ha ha1 hk L hL q hq
have hqcb (n : ℕ) : ‖fourierConjugate (q n)‖ ≤ M := by simpa only [fourierConjugate_norm] using hqb n
have hqcp := ExpandingCompactApproximation.positivePower a k M ha ha1 hk L hL hLinf
(fun n => fourierConjugate (q n)) hqc hqcb m
let B₁ := expandingAlgebraBound a k ^ (m + 2) * M ^ (m + 2 + 1)
let B₂ := expandingAlgebraBound a k ^ m * M ^ (m + 1)
have hb₁ (n : ℕ) : ‖expandingPower a k (L n) ha ha1 hk (hL n) (q n) (m + 2 + 1)‖ ≤ B₁ :=
(expandingPower_positive_norm_le a k (L n) ha ha1 hk (hL n) (q n) (m + 2)).trans
(mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) (hqb n) _)
(pow_nonneg (expandingAlgebraBound_nonneg a k) _))
have hb₂ (n : ℕ) : ‖expandingPower a k (L n) ha ha1 hk (hL n) (fourierConjugate (q n)) (m + 1)‖ ≤ B₂ :=
(expandingPower_positive_norm_le a k (L n) ha ha1 hk (hL n) (fourierConjugate (q n)) m).trans
(mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) (hqcb n) _)
(pow_nonneg (expandingAlgebraBound_nonneg a k) _))
have hp := ExpandingCompactApproximation.mul a k (max B₁ B₂) ha ha1 hk L hL hLinf _ _ hqp hqcp
(fun n => (hb₁ n).trans (le_max_left _ _)) (fun n => (hb₂ n).trans (le_max_right _ _))
have hs := ExpandingCompactApproximation.smul a k ha1 hk L hL _ hp ((m + 2 : ℕ) : ℂ)
have he : m + 2 - 1 = m + 1 := by omega
simpa only [expandingAnticircularCoefficient, he] using hs
end DefocusingNLS
end OAI
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