File size: 3,093 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 52 53 54 55 56 57 58 59 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingSampleDerivativeFormula
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactFilterLimit
namespace OAI
/-! # Low-frequency derivative compactness for the actual sampled product -/
open Filter Topology
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem tendsto_expandingSample_low_derivative (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) (j : Fin N → Fin 12) :
Tendsto (fun n => expandingOrderedFourierEnergy a (L n) N (hL n) j
(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 G := fun s : Finset (Fin N) => schwartzPartialDerivative N j s K
let Q := fun n (s : Finset (Fin N)) =>
schwartzTorusSample a N (L n) ha1 hN (hL n) (radianFourierKernel (G s))
let F := fun n (s : Finset (Fin N)) =>
expandingSchwartzFilter (L n) (partialFilterKernel S hS N j sᶜ) (f n)
let V := fun n (s : Finset (Fin N)) => expandingPhysicalMassVector a N (L n) ha ha1 hN (hL n)
(expandingProduct a N (L n) ha ha1 hN (hL n) (Q n s) (F n s))
have hv (s : Finset (Fin N)) : Tendsto (fun n => V n s) atTop (𝓝 0) := by
rw [tendsto_zero_iff_norm_tendsto_zero]
have ht := tendsto_expandingCompactPhysical_filter a N M R ha ha1 hN L hL hLinf f hf hlocal
(partialFilterKernel S hS N j sᶜ) (G s) (schwartzPartialDerivative_zero N j s K R hK)
have he (n : ℕ) : ‖V n s‖ = ‖torusPhysicalL2Value (L n)
(expandingUnitTorusFunction a N (L n) (Q n s) *
expandingUnitTorusFunction a N (L n) (F n s))‖ := by
rw [← LinearIsometryEquiv.norm_map torusFourierIsometry (V n s)]
dsimp only [V]
rw [expandingPhysicalMassVector_isometry, expandingUnitTorusFunction_product]
change Tendsto (fun n => ‖torusPhysicalL2Value (L n)
(expandingUnitTorusFunction a N (L n) (Q n s) *
expandingUnitTorusFunction a N (L n) (F n s))‖) atTop (𝓝 0) at ht
simpa only [he] using ht
have hs := tendsto_finsetSum Finset.univ (fun s _ => hv s)
have he (n : ℕ) : expandingOrderedFourierEnergy a (L n) N (hL n) j
(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))) = ∑ s : Finset (Fin N), V n s :=
expandingSample_low_derivative_formula a (L n) S N ha ha1 hN (hL n) hS j K (f n)
simp_rw [← he] at hs
simpa only [Finset.sum_const_zero] using hs
end DefocusingNLS
end OAI
|