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9eeb9c7 | 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 38 39 40 41 42 43 44 45 46 47 48 49 50 | 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
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