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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 33 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingApproximationProduct
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingPositivePower
namespace OAI
/-! # Positive powers preserve compact coefficient approximation -/
open Filter Topology
namespace DefocusingNLS
theorem ExpandingCompactApproximation.positivePower (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) (p : ℕ) :
ExpandingCompactApproximation a k ha1 hk L hL
(fun n => expandingPower a k (L n) ha ha1 hk (hL n) (q n) (p + 1)) := by
induction p with
| zero => simpa only [zero_add, expandingPower_one] using hq
| succ p ih =>
let B := expandingAlgebraBound a k ^ p * M ^ (p + 1)
have hpb (n : ℕ) : ‖expandingPower a k (L n) ha ha1 hk (hL n) (q n) (p + 1)‖ ≤ B :=
(expandingPower_positive_norm_le a k (L n) ha ha1 hk (hL n) (q n) p).trans
(mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) (hqb n) (p + 1))
(pow_nonneg (expandingAlgebraBound_nonneg a k) p))
exact ExpandingCompactApproximation.mul a k (max M B) ha ha1 hk L hL hLinf q _ hq ih
(fun n => (hqb n).trans (le_max_left _ _)) (fun n => (hpb n).trans (le_max_right _ _))
end DefocusingNLS
end OAI
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