openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingApproximationCoefficients.lean
| 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 | |