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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | 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
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