| import OAI.Analysis.AncientFlat.Spectral.ResolventCoefficients | |
| import OAI.Analysis.AncientFlat.Kernel.GreenConvolution | |
| namespace OAI | |
| noncomputable section | |
| open scoped Topology BoundedContinuousFunction | |
| namespace FlatRecursion | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] | |
| theorem exists_bounded_recursion_inverse (P : E →L[ℂ] E) (q : ℝ) (hq : 0<q) | |
| (hcircle : ∀ z : ℂ, ‖z‖=q → z ∈ resolventSet ℂ P) : | |
| ∃ D : (ℤ →ᵇ E) →L[ℂ] (ℤ →ᵇ E), ∀ f : ℤ →ᵇ E, ∀ j : ℤ, | |
| (q : ℂ) • D f (j+1)=P (D f j)+f j := by | |
| let G := PeriodicResolvent.coeff (PeriodicResolvent.resolventLoop P q) | |
| exact ⟨greenOperator G, greenOperator_solves P q G | |
| (PeriodicResolvent.green_summable P hq hcircle) | |
| (PeriodicResolvent.green_identity P hq hcircle)⟩ | |
| end FlatRecursion | |
| end | |
| end OAI | |