The continuous Fourier representative has the prescribed pointwise initial trace.

theorem AVenhance.Infra.Classical.classicalInitialPointwiseOneLeTwo : Fact (1 ≤ 2)

@[instance_reducible]

noncomputable def AVenhance.Infra.Classical.classicalInitialPointwiseMeasureSpace : MeasureTheory.MeasureSpace UnitAddCircle

Use normalized Haar measure on the unit circle for periodic integrals.

Equations

• AVenhance.Infra.Classical.classicalInitialPointwiseMeasureSpace = { toMeasurableSpace := QuotientAddGroup.measurableSpace (AddSubgroup.zmultiples 1), volume := AddCircle.haarAddCircle }

theorem AVenhance.Infra.Classical.classicalInitialPointwiseAddHaar : MeasureTheory.volume.IsAddHaarMeasure

theorem AVenhance.Infra.Classical.classicalInitialPointwiseProbability : MeasureTheory.IsProbabilityMeasure MeasureTheory.volume

theorem AVenhance.Infra.Classical.classicalGalerkinRealSmoothLift_initial (φ : ℝ → Homogenization.Vec 2 → ℝ) (hφ : IsAdmissibleStream φ) (κ : ℝ) (hκ : 0 < κ) (F : ℝ → Homogenization.Vec 2 → ℝ) (hF : ContDiffOn ℝ (↑⊤) (Function.uncurry F) (Set.Ici 0 ×ˢ Set.univ)) (hFper : ∀ (t : ℝ), 0 ≤ t → IsZ2Periodic (F t)) (θ₀ : Homogenization.Vec 2 → ℝ) (hθ₀ : ContDiff ℝ (↑⊤) θ₀) (hθ₀per : IsZ2Periodic θ₀) (x : Homogenization.Vec 2) : classicalGalerkinRealSmoothLift φ hφ κ hκ F hF hFper θ₀ hθ₀ hθ₀per ⟨0, ⋯⟩ x = θ₀ x

The real smooth representative agrees pointwise with the datum at time zero.