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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.TorusCompactWeightIntegral
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.UnitTorusPhysical
namespace OAI
/-! # Physical L² identity for the actual sampled torus coefficient -/
open MeasureTheory
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "T" => UnitAddTorus (Fin 12)
noncomputable local instance expandingCompactWeightMeasure : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩
local instance expandingCompactWeightProbability : IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
theorem expandingCompactWeight_integral (a k L R : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L)
(hLR : 2 * R < 2 * Real.pi * L) (K : 𝓢(E, ℂ))
(hK : ∀ y : E, R < ‖y‖ → K y = 0) (f : FourierL2) :
(2 * Real.pi * L) ^ 12 * (∫ x : T,
‖expandingUnitTorusFunction a k L
(schwartzTorusSample a k L ha1 hk hL (radianFourierKernel K)) x *
expandingUnitTorusFunction a k L f x‖ ^ 2) =
∫ y : E, ‖K y * expandingPhysicalContinuous a k L ha ha1 hk hL f y‖ ^ 2 := by
rw [expandingUnitTorusFunction_sample a k L ha ha1 hk hL]
rw [separatedSchwartz_torus_weighted_integral L R (by linarith) hLR K hK]
apply integral_congr_ae
filter_upwards [] with y
rw [expandingUnitTorusFunction_physical a k L ha ha1 hk hL]
end DefocusingNLS
end OAI
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