| import OAI.Analysis.IrrationalRotation.Model.WeightedShift |
| import Mathlib.MeasureTheory.Function.L2Space |
| import Mathlib.LinearAlgebra.LinearPMap |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal |
| namespace Invariant233.WeightedShift |
| variable {X : Type*} [MeasurableSpace X] (μ : Measure X) |
|
|
| abbrev FieldHilbert := Lp Hilbert 2 μ |
|
|
| def fieldMultiplierGraph (h : X → ℤ → ℂ) : |
| Submodule ℂ (FieldHilbert μ × FieldHilbert μ) where |
| carrier := {p | ∀ᵐ x ∂μ, ∀ k, p.2 x k = h x k * p.1 x k} |
| zero_mem' := by |
| filter_upwards [Lp.coeFn_zero Hilbert 2 μ] with x hx |
| intro k |
| simp only [Prod.fst_zero, Prod.snd_zero, hx, Pi.zero_apply, lp.coeFn_zero, mul_zero] |
| add_mem' := by |
| intro p q hp hq |
| filter_upwards [hp, hq, Lp.coeFn_add p.1 q.1, Lp.coeFn_add p.2 q.2] |
| with x hp hq hfst hsnd |
| intro k |
| change (p.2 + q.2) x k = h x k * (p.1 + q.1) x k |
| rw [hsnd, hfst] |
| change p.2 x k + q.2 x k = h x k * (p.1 x k + q.1 x k) |
| rw [hp, hq, mul_add] |
| smul_mem' := by |
| intro c p hp |
| filter_upwards [hp, Lp.coeFn_smul c p.1, Lp.coeFn_smul c p.2] |
| with x hp hfst hsnd |
| intro k |
| change (c • p.2) x k = h x k * (c • p.1) x k |
| rw [hsnd, hfst] |
| change c * p.2 x k = h x k * (c * p.1 x k) |
| rw [hp, mul_left_comm] |
|
|
| def fieldUnboundedMultiplier (h : X → ℤ → ℂ) : |
| FieldHilbert μ →ₗ.[ℂ] FieldHilbert μ := |
| (fieldMultiplierGraph μ h).toLinearPMap |
|
|
| end Invariant233.WeightedShift |
|
|
| end |
|
|
| end OAI |
|
|
