| import Mathlib |
| import OAI.Analysis.BrownMeasure.Modulus |
| import OAI.Analysis.FiniteFactor.Matrix.Covariance |
| import OAI.Analysis.FiniteFactor.Matrix.Integrals |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter InnerProductSpace |
| open scoped ComplexInnerProductSpace Topology |
| namespace FiniteFactor |
| local instance : IsometricContinuousFunctionalCalculus ℝ ProductOp IsSelfAdjoint := |
| IsSelfAdjoint.instIsometricContinuousFunctionalCalculus |
| abbrev WeightInterval := Set.Icc (0:ℝ) 1 |
|
|
| def sourceWeightPoint (θ : SourceCoordinate → ℝ) (x : SourceCoordinate → ℝ) : WeightInterval := |
| ⟨min 1 (sourceCocycle θ sourceLogWeight 1 x),le_min zero_le_one (sourceCocycle_pos _ _ _ _).le,min_le_left _ _⟩ |
| lemma sourceWeightPoint_measurable (θ : SourceCoordinate → ℝ) : Measurable (sourceWeightPoint θ) := by |
| apply Measurable.subtype_mk |
| exact measurable_const.min (sourceCocycle_measurable θ sourceLogWeight 1) |
| lemma sourceWeightPoint_val (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) |
| {x : SourceCoordinate → ℝ} (hx : ∀ i,x i∈Set.Ico 0 1) : |
| (sourceWeightPoint θ x:ℝ)=sourceCocycle θ sourceLogWeight 1 x := |
| min_eq_right (sourceWeight_le_one θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg hx) |
|
|
| def sourceContinuousWeightBase (θ : SourceCoordinate → ℝ) (φ : C(WeightInterval,ℝ)) : SourceBaseFunction where |
| toFun x := (φ (sourceWeightPoint θ x):ℂ) |
| measurable := Complex.measurable_ofReal.comp (φ.continuous.measurable.comp (sourceWeightPoint_measurable θ)) |
| bound := ‖φ‖ |
| bound_nonneg := norm_nonneg _ |
| norm_le x _ := by simpa only [Complex.norm_real] using φ.norm_coe_le_norm (sourceWeightPoint θ x) |
|
|
| def sourceContinuousWeight (θ : SourceCoordinate → ℝ) (φ : C(WeightInterval,ℝ)) : ProductOp := |
| sourcePi θ (sourceContinuousWeightBase θ φ) |
|
|
| lemma sourceContinuousWeight_apply_ae (θ : SourceCoordinate → ℝ) (φ : C(WeightInterval,ℝ)) |
| (u : SourceHilbert SourceCoordinate) : |
| sourceContinuousWeight θ φ u=ᵐ[sourceMeasure SourceCoordinate] fun z=> |
| (φ (sourceWeightPoint θ (coordinateRotate θ z.2 z.1)):ℂ)*u z := |
| sourcePi_apply_ae θ (sourceContinuousWeightBase θ φ) u |
|
|
| lemma sourceContinuousWeight_one (θ : SourceCoordinate → ℝ) : sourceContinuousWeight θ 1=1 := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourceContinuousWeight_apply_ae θ 1 u] with z hz |
| change (sourceContinuousWeight θ 1 u) z=u z |
| simpa using hz |
| lemma sourceContinuousWeight_zero (θ : SourceCoordinate → ℝ) : sourceContinuousWeight θ 0=0 := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourceContinuousWeight_apply_ae θ 0 u,Lp.coeFn_zero ℂ 2 (sourceMeasure SourceCoordinate)] with z hz h0 |
| change (sourceContinuousWeight θ 0 u) z=(0:SourceHilbert SourceCoordinate) z |
| rw [h0] |
| simpa using hz |
| lemma sourceContinuousWeight_add (θ : SourceCoordinate → ℝ) (φ ψ : C(WeightInterval,ℝ)) : |
| sourceContinuousWeight θ (φ+ψ)=sourceContinuousWeight θ φ+sourceContinuousWeight θ ψ := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourceContinuousWeight_apply_ae θ (φ+ψ) u,sourceContinuousWeight_apply_ae θ φ u, |
| sourceContinuousWeight_apply_ae θ ψ u,Lp.coeFn_add (sourceContinuousWeight θ φ u) (sourceContinuousWeight θ ψ u)] with z ha hf hg hs |
| change (sourceContinuousWeight θ (φ+ψ) u) z=(sourceContinuousWeight θ φ u+sourceContinuousWeight θ ψ u) z |
| rw [ha,hs,Pi.add_apply,hf,hg] |
| simp [add_mul] |
| lemma sourceContinuousWeight_mul (θ : SourceCoordinate → ℝ) (φ ψ : C(WeightInterval,ℝ)) : |
| sourceContinuousWeight θ (φ*ψ)=sourceContinuousWeight θ φ*sourceContinuousWeight θ ψ := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourceContinuousWeight_apply_ae θ (φ*ψ) u, |
| sourceContinuousWeight_apply_ae θ φ (sourceContinuousWeight θ ψ u),sourceContinuousWeight_apply_ae θ ψ u] with z ha hf hg |
| change (sourceContinuousWeight θ (φ*ψ) u) z=(sourceContinuousWeight θ φ (sourceContinuousWeight θ ψ u)) z |
| rw [ha,hf,hg] |
| simp [mul_assoc] |
| lemma sourceContinuousWeight_smul (θ : SourceCoordinate → ℝ) (r : ℝ) (φ : C(WeightInterval,ℝ)) : |
| sourceContinuousWeight θ (r • φ)=r • sourceContinuousWeight θ φ := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourceContinuousWeight_apply_ae θ (r • φ) u,sourceContinuousWeight_apply_ae θ φ u, |
| Lp.coeFn_smul r (sourceContinuousWeight θ φ u)] with z ha hf hs |
| change (sourceContinuousWeight θ (r • φ) u) z=(r • sourceContinuousWeight θ φ u) z |
| rw [ha,hs,Pi.smul_apply,hf] |
| simp [Complex.real_smul,mul_assoc] |
| lemma sourceContinuousWeight_star (θ : SourceCoordinate → ℝ) (φ : C(WeightInterval,ℝ)) : |
| sourceContinuousWeight θ (star φ)=star (sourceContinuousWeight θ φ) := by |
| change sourcePi θ (sourceContinuousWeightBase θ (star φ))=star (sourcePi θ (sourceContinuousWeightBase θ φ)) |
| rw [sourcePi_star] |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourcePi_apply_ae θ (sourceContinuousWeightBase θ (star φ)) u, |
| sourcePi_apply_ae θ (sourceBaseStar (sourceContinuousWeightBase θ φ)) u] with z h1 h2 |
| rw [h1,h2] |
| simp [sourcePiSymbol,sourceContinuousWeightBase,sourceBaseStar] |
|
|
| def sourceContinuousWeightHom (θ : SourceCoordinate → ℝ) : C(WeightInterval,ℝ) →⋆ₐ[ℝ] ProductOp where |
| toFun := sourceContinuousWeight θ |
| map_one' := sourceContinuousWeight_one θ |
| map_zero' := sourceContinuousWeight_zero θ |
| map_add' := sourceContinuousWeight_add θ |
| map_mul' := sourceContinuousWeight_mul θ |
| commutes' r := by |
| rw [Algebra.algebraMap_eq_smul_one,sourceContinuousWeight_smul,sourceContinuousWeight_one,Algebra.algebraMap_eq_smul_one] |
| map_star' := sourceContinuousWeight_star θ |
|
|
| lemma sourceContinuousWeightHom_continuous (θ : SourceCoordinate → ℝ) : Continuous (sourceContinuousWeightHom θ) := by |
| exact ((sourceContinuousWeightHom θ).toAlgHom.toLinearMap.mkContinuous (1:ℝ) |
| (fun φ=>by |
| change ‖sourcePi θ (sourceContinuousWeightBase θ φ)‖ ≤ 1*‖φ‖ |
| simpa only [one_mul, sourceContinuousWeightBase] using sourcePi_norm_le θ (sourceContinuousWeightBase θ φ))).continuous |
|
|
| lemma sourceContinuousWeight_id (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) : |
| sourceContinuousWeight θ ((ContinuousMap.id ℝ).restrict WeightInterval)=sourcePi θ (sourceWeightBase θ hθ) := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourceContinuousWeight_apply_ae θ ((ContinuousMap.id ℝ).restrict WeightInterval) u, |
| sourcePi_apply_ae θ (sourceWeightBase θ hθ) u] with z h1 h2 |
| change (sourceContinuousWeight θ _ u) z=(sourcePi θ (sourceWeightBase θ hθ) u) z |
| rw [h1,h2] |
| simp only [ContinuousMap.restrict_apply,ContinuousMap.id_apply, |
| sourcePiSymbol,sourceWeightBase] |
| have hc : ∀ i,coordinateRotate θ z.2 z.1 i∈Set.Ico (0:ℝ) 1 := by |
| intro i |
| exact ⟨Int.fract_nonneg _,Int.fract_lt_one _⟩ |
| rw [sourceWeightPoint_val θ hθ hc] |
|
|
| lemma sourceWeight_spectrum_subset (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) : |
| spectrum ℝ (sourcePi θ (sourceWeightBase θ hθ))⊆WeightInterval := by |
| intro t ht |
| exact ⟨spectrum_nonneg_of_nonneg (sourceWeight_positive θ hθ) ht, |
| (Real.le_norm_self t).trans ((spectrum.norm_le_norm_of_mem ht).trans (sourcePi_norm_le θ (sourceWeightBase θ hθ)))⟩ |
|
|
| lemma sourceContinuousWeightHom_eq_cfc (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) : |
| sourceContinuousWeightHom θ=cfcHomSuperset (sourceWeight_positive θ hθ).isSelfAdjoint (sourceWeight_spectrum_subset θ hθ) := by |
| apply ContinuousMap.UniqueHom.eq_of_continuous_of_map_id |
| · exact sourceContinuousWeightHom_continuous θ |
| · exact cfcHomSuperset_continuous _ _ |
| · rw [cfcHomSuperset_id] |
| exact sourceContinuousWeight_id θ hθ |
|
|
| lemma sourcePi_trace (θ : SourceCoordinate → ℝ) (g : SourceBaseFunction) : |
| sourceVacuumFunctional (sourcePi θ g)=∫ x,g.toFun x ∂sourceBaseMeasure := by |
| rw [sourceVacuumFunctional_matrixIntegral] |
| apply integral_congr_ae |
| filter_upwards [sourceMatrixEntry_pi θ g 0,sourceSamplingHaar_cube] with x hx hc |
| have hr : coordinateRotate θ 0 x=x := by |
| funext i |
| simpa [coordinateRotate] using Int.fract_eq_self.mpr (hc i) |
| simpa only [hr,ite_true] using hx 0 |
| end FiniteFactor |
| end |
|
|
| end OAI |
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|
