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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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingApproximationLinear
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProductDifference
namespace OAI
/-! # Compact coefficient approximation is preserved by the actual torus product -/
open Filter Topology
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem ExpandingCompactApproximation.mul (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 r : ℕ → FourierL2) (hq : ExpandingCompactApproximation a k ha1 hk L hL q)
(hr : ExpandingCompactApproximation a k ha1 hk L hL r)
(hqb : ∀ n, ‖q n‖ ≤ M) (hrb : ∀ n, ‖r n‖ ≤ M) :
ExpandingCompactApproximation a k ha1 hk L hL
(fun n => expandingProduct a k (L n) ha ha1 hk (hL n) (q n) (r n)) := by
have hM : 0 ≤ M := (norm_nonneg (q 0)).trans (hqb 0)
let A := expandingAlgebraBound a k
have hA : 0 ≤ A := expandingAlgebraBound_nonneg a k
let C := A * (2 * M + 1)
have hC : 0 ≤ C := by dsimp [C]; positivity
intro ε hε
let δ := min 1 (ε / (C + 1))
have hδ : 0 < δ := lt_min (by norm_num) (div_pos hε (by positivity))
have hδ1 : δ ≤ 1 := min_le_left _ _
have hδε : δ * (C + 1) ≤ ε := (le_div_iff₀ (by positivity : 0 < C + 1)).mp (min_le_right _ _)
obtain ⟨R, K, hK, heq⟩ := hq δ hδ
obtain ⟨S, J, hJ, her⟩ := hr δ hδ
let T := max R S
have hKT : ∀ y : E, T < ‖y‖ → K y = 0 := fun y hy => hK y ((le_max_left R S).trans_lt hy)
have hJT : ∀ y : E, T < ‖y‖ → J y = 0 := fun y hy => hJ y ((le_max_right R S).trans_lt hy)
refine ⟨T, SchwartzMap.smulLeftCLM ℂ K J, ?_, ?_⟩
· intro y hy
rw [SchwartzMap.smulLeftCLM_apply_apply K.hasTemperateGrowth, hKT y hy, zero_smul]
· filter_upwards [heq, her, hLinf.eventually (eventually_ge_atTop T)] with n hnq hnr hnL
have hsep : 2 * T < 2 * Real.pi * L n := calc
2 * T ≤ 2 * L n := mul_le_mul_of_nonneg_left hnL (by norm_num)
_ < 2 * Real.pi * L n := mul_lt_mul_of_pos_right
(by linarith [Real.pi_gt_three] : (2 : ℝ) < 2 * Real.pi) (by linarith [hL n])
rw [← schwartzTorusSample_product a k (L n) T ha ha1 hk (hL n) hsep K J hKT hJT]
have hKB : ‖schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K)‖ ≤ M + 1 := by
calc
_ ≤ ‖schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K) - q n‖ + ‖q n‖ :=
norm_le_norm_sub_add _ _
_ ≤ δ + M := add_le_add (by rw [norm_sub_rev]; exact hnq.le) (hqb n)
_ ≤ M + 1 := by linarith
have hb := expandingProduct_difference_norm a k (L n) ha ha1 hk (hL n) (q n) (r n)
(schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K))
(schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel J))
calc
_ ≤ A * (‖q n - schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K)‖ * ‖r n‖ +
‖schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K)‖ *
‖r n - schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel J)‖) := hb
_ ≤ A * (δ * M + (M + 1) * δ) :=
mul_le_mul_of_nonneg_left (add_le_add
(mul_le_mul hnq.le (hrb n) (norm_nonneg _) hδ.le)
(mul_le_mul hKB hnr.le (norm_nonneg _) (by linarith))) hA
_ = C * δ := by dsimp [C]; ring
_ < ε := by nlinarith
end DefocusingNLS
end OAI
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