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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 34 35 36 37 38 39 40 41 42 43 44 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactWeightLimit
import OAI.MathematicalPhysics.DefocusingNLS.Linear.TorusPhysicalL2
namespace OAI
/-! # Norm compactness of the sampled coefficient times filtered derivatives -/
open Filter Topology MeasureTheory
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "T" => UnitAddTorus (Fin 12)
noncomputable local instance expandingCompactPhysicalMeasure : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩
local instance expandingCompactPhysicalProbability : IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
theorem tendsto_expandingCompactPhysical_derivative (a k M R S : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hS : 0 < S)
(N : ℕ) (j : Fin N → Fin 12) (L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n)
(hLinf : Tendsto L atTop atTop) (f : ℕ → FourierL2) (hf : ∀ n, ‖f n‖ ≤ M)
(hlocal : ∀ R ε : ℝ, 0 < ε → ∀ᶠ n in atTop, ∀ y : E, ‖y‖ ≤ R →
‖expandingTorusFunction a k (L n) (f n) (euclideanToTorus ((L n)⁻¹ • y))‖ < ε)
(K : 𝓢(E, ℂ)) (hK : ∀ y : E, R < ‖y‖ → K y = 0) :
Tendsto (fun n => ‖torusPhysicalL2Value (L n)
(expandingUnitTorusFunction a k (L n)
(schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K)) *
expandingUnitTorusFunction a k (L n)
(expandingSmoothDerivative (L n) S hS N j (f n)))‖) atTop (𝓝 0) := by
have ht := tendsto_expandingCompactWeight_derivative a k M R S ha ha1 hk hS N j
L hL hLinf f hf hlocal K hK
have hs : Tendsto (fun n => ‖torusPhysicalL2Value (L n)
(expandingUnitTorusFunction a k (L n)
(schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K)) *
expandingUnitTorusFunction a k (L n)
(expandingSmoothDerivative (L n) S hS N j (f n)))‖ ^ 2) atTop (𝓝 0) := by
simpa only [torusPhysicalL2Value_norm_sq, ContinuousMap.mul_apply] using ht
simpa only [Real.sqrt_sq (norm_nonneg _), Real.sqrt_zero] using hs.sqrt
end DefocusingNLS
end OAI
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