File size: 2,441 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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactDerivativeLimit
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingMassDerivativeNorm
namespace OAI
/-! # Strong compactness of a sampled coefficient after smooth frequency cutoff -/
open Filter Topology
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem tendsto_expandingSample_low_product (a M R S : ℝ) (N : ℕ)
(ha : 0 < a) (ha1 : a < 1) (hN : 8 < (N : ℝ)) (hS : 0 < S)
(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 N (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 => expandingProduct a N (L n) ha ha1 hN (hL n)
(schwartzTorusSample a N (L n) ha1 hN (hL n) (radianFourierKernel K))
(expandingSmoothLow (L n) S hS (f n))) atTop (𝓝 0) := by
let q := fun n => schwartzTorusSample a N (L n) ha1 hN (hL n) (radianFourierKernel K)
let u := fun n => expandingSmoothLow (L n) S hS (f n)
let g := fun n => expandingProduct a N (L n) ha ha1 hN (hL n) (q n) (u n)
apply tendsto_expanding_of_mass_derivatives_zero a N ha ha1 hN L hL g
· have ht := tendsto_expandingCompactPhysical_filter a N M R ha ha1 hN L hL hLinf f hf hlocal
(radianInverseKernel (smoothFrequencyCutoff S hS)) K hK
have he (n : ℕ) : ‖expandingPhysicalMassVector a N (L n) ha ha1 hN (hL n) (g n)‖ =
‖torusPhysicalL2Value (L n) (expandingUnitTorusFunction a N (L n) (q n) *
expandingUnitTorusFunction a N (L n)
(expandingSchwartzFilter (L n) (radianInverseKernel (smoothFrequencyCutoff S hS)) (f n)))‖ := by
rw [← LinearIsometryEquiv.norm_map torusFourierIsometry
(expandingPhysicalMassVector a N (L n) ha ha1 hN (hL n) (g n)),
expandingPhysicalMassVector_isometry]
dsimp only [g]
rw [expandingUnitTorusFunction_product]
dsimp only [u]
rw [expandingSmoothLow_eq_convolution]
simpa only [he] using ht
· intro j
have ht := tendsto_expandingSample_low_derivative a M R S N ha ha1 hN hS
L hL hLinf f hf hlocal K hK j
simpa only [norm_zero] using ht.norm
end DefocusingNLS
end OAI
|