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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingApproximationCircular
namespace OAI
/-! # Both actual odd-power coefficients have compact approximations -/
open Filter Topology
namespace DefocusingNLS
theorem expandingOdd_coefficients_compact (a k Q : 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‖ ≤ Q) (m : ℕ) (hm : 0 < m) :
ExpandingCompactApproximation a k ha1 hk L hL
(fun n => expandingCircularCoefficient a k (L n) ha ha1 hk (hL n) m (q n)) ∧
ExpandingCompactApproximation a k ha1 hk L hL
(fun n => expandingAnticircularCoefficient a k (L n) ha ha1 hk (hL n) m (q n)) := by
cases m with
| zero => omega
| succ m =>
constructor
· exact ExpandingCompactApproximation.circular a k Q ha ha1 hk L hL hLinf q hq hqb m
· cases m with
| zero => exact ExpandingCompactApproximation.anticircular_one a k Q ha ha1 hk L hL hLinf q hq hqb
| succ m => exact ExpandingCompactApproximation.anticircular_succ a k Q ha ha1 hk L hL hLinf q hq hqb m
end DefocusingNLS
end OAI
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