| import OAI.Analysis.FiniteFactor.CrossedProduct.FiniteVectors |
| import Mathlib.Topology.Instances.Complex |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped ENNReal |
| noncomputable section |
|
|
| /-- Continuity in a separable parameter lets a countable dense family of |
| almost-everywhere inequalities determine every parameter on one conull set. -/ |
| theorem ae_forall_le_of_continuous {X E : Type*} [MeasurableSpace X] |
| (μ : Measure X) [TopologicalSpace E] [TopologicalSpace.SeparableSpace E] |
| (f g : E → X → ℝ) (hf : ∀ x, Continuous (fun e => f e x)) |
| (hg : ∀ x, Continuous (fun e => g e x)) |
| (h : ∀ e, ∀ᵐ x ∂μ, f e x ≤ g e x) : |
| ∀ᵐ x ∂μ, ∀ e, f e x ≤ g e x := by |
| obtain ⟨s, hsc, hsd⟩ := TopologicalSpace.exists_countable_dense E |
| have hs : ∀ᵐ x ∂μ, ∀ e ∈ s, f e x ≤ g e x := |
| (ae_ball_iff hsc).mpr fun e _ => h e |
| filter_upwards [hs] with x hx e |
| exact le_on_closure hx (hf x).continuousOn (hg x).continuousOn (hsd e) |
|
|
| theorem sourceFiberEnergy_finite_le (f : SourceHilbert SourceCoordinate) (s : Finset ℤ) : |
| ∀ᵐ x ∂sourceBaseMeasure, ∑ n ∈ s, ‖f (x,n)‖ ^ 2 ≤ sourceFiberEnergy f x := by |
| filter_upwards [(source_norm_sq_integrable f).prod_right_ae] with x hx |
| have h := setIntegral_le_integral (μ := Measure.count) (s := (s : Set ℤ)) |
| hx (Eventually.of_forall fun n => sq_nonneg ‖f (x,n)‖) |
| rw [setIntegral_finset s hx.integrableOn] at h |
| simpa [measureReal_def, sourceFiberEnergy] using h |
|
|
| /-- Each finite scalar matrix compression satisfies the actual operator norm bound. -/ |
| theorem sourceMatrix_finite_bound (θ : SourceCoordinate → ℝ) |
| {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate} |
| (hH : H ∈ sourceAlgebra θ) (s t : Finset ℤ) (a : ℤ → ℂ) : |
| ∀ᵐ x ∂sourceBaseMeasure, |
| (∑ k ∈ t, ‖∑ n ∈ s, a n * sourceMatrixEntry H k n x‖ ^ 2) ≤ |
| ‖H‖ ^ 2 * ∑ n ∈ s, ‖a n‖ ^ 2 := by |
| filter_upwards [sourceFiniteVector_image_coe H s a, |
| sourceFiniteVector_energy s a, |
| sourceFiberEnergy_finite_le (H (sourceFiniteVector s a)) t, |
| sourceAlgebra_fiberEnergy_le θ hH (sourceFiniteVector s a)] with x hc he hf hh |
| calc |
| _ = ∑ k ∈ t, ‖(H (sourceFiniteVector s a)) (x,k)‖ ^ 2 := by simp_rw [hc] |
| _ ≤ sourceFiberEnergy (H (sourceFiniteVector s a)) x := hf |
| _ ≤ ‖H‖ ^ 2 * sourceFiberEnergy (sourceFiniteVector s a) x := hh |
| _ = _ := by rw [he] |
|
|
| /-- One conull set works for every complex finite input vector and every finite |
| output truncation. No field or boundedness hypothesis has been substituted. -/ |
| theorem sourceMatrix_finite_bound_all (θ : SourceCoordinate → ℝ) |
| {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate} |
| (hH : H ∈ sourceAlgebra θ) : |
| ∀ᵐ x ∂sourceBaseMeasure, ∀ (s t : Finset ℤ) (a : ℤ → ℂ), |
| (∑ k ∈ t, ‖∑ n ∈ s, a n * sourceMatrixEntry H k n x‖ ^ 2) ≤ |
| ‖H‖ ^ 2 * ∑ n ∈ s, ‖a n‖ ^ 2 := by |
| rw [ae_all_iff] |
| intro s |
| rw [ae_all_iff] |
| intro t |
| apply ae_forall_le_of_continuous (E := ℤ → ℂ) sourceBaseMeasure |
| (fun a x => ∑ k ∈ t, ‖∑ n ∈ s, a n * sourceMatrixEntry H k n x‖ ^ 2) |
| (fun a _ => ‖H‖ ^ 2 * ∑ n ∈ s, ‖a n‖ ^ 2) |
| · intro x; fun_prop |
| · intro x; fun_prop |
| · exact sourceMatrix_finite_bound θ hH s t |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
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|
