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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 45 46 47 48 49 50 51 52 53 54 55 56 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCoefficientCompactness
import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzProductSampling
namespace OAI
/-! # Scalar multiplication and conjugation preserve compact coefficient approximation -/
open Filter Topology
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
variable (a k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k)
(L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (q : ℕ → FourierL2)
theorem ExpandingCompactApproximation.smul
(hq : ExpandingCompactApproximation a k ha1 hk L hL q) (c : ℂ) :
ExpandingCompactApproximation a k ha1 hk L hL (fun n => c • q n) := by
intro ε hε
let δ := ε / (‖c‖ + 1)
have hδ : 0 < δ := div_pos hε (by positivity)
obtain ⟨R, K, hK, herr⟩ := hq δ hδ
refine ⟨R, c • K, ?_, ?_⟩
· intro y hy
simp only [smul_apply, hK y hy, smul_zero]
· have hbound : ‖c‖ * δ < ε := by
have hi : δ * (‖c‖ + 1) = ε := div_mul_cancel₀ _ (by positivity)
nlinarith
filter_upwards [herr] with n hn
have he : schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel (c • K)) =
c • schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K) :=
(physicalSchwartzTorusSamplingLinear a k (L n) ha1 hk (hL n)).map_smul c K
rw [he, ← smul_sub, norm_smul]
exact (mul_le_mul_of_nonneg_left hn.le (norm_nonneg c)).trans_lt hbound
include ha in
theorem ExpandingCompactApproximation.conjugate
(hq : ExpandingCompactApproximation a k ha1 hk L hL q) :
ExpandingCompactApproximation a k ha1 hk L hL (fun n => fourierConjugate (q n)) := by
intro ε hε
obtain ⟨R, K, hK, herr⟩ := hq ε hε
refine ⟨R, schwartzPointConjugate K, ?_, ?_⟩
· intro y hy
simp only [schwartzPointConjugate_apply, hK y hy, map_zero]
· filter_upwards [herr] with n hn
rw [← schwartzTorusSample_conjugate a k (L n) ha ha1 hk (hL n)]
change ‖sobolevConjugation (q n) - sobolevConjugation _‖ < ε
rw [← map_sub, sobolevConjugation_apply, fourierConjugate_norm]
exact hn
end DefocusingNLS
end OAI
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