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b224933 | 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 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Shells.Shells
import OAI.MeasureTheory.FalconerCompletion.Fourier.FourierPolynomialSmoothing
namespace OAI
/-! Dyadic shells and prefix decompositions. -/
open _root_.OAI.PlanarFalconer
section
section
namespace CurrentFalconer.MonotoneShell
open MeasureTheory Set Filter
open scoped Topology ENNReal
noncomputable def complexOfPositive (μ : FiniteMeasure ℝ) : ComplexMeasure ℝ :=
(μ : Measure ℝ).toSignedMeasure.toComplexMeasure 0
noncomputable def complexFourier (ν : ComplexMeasure ℝ) (r : ℝ) : ℂ :=
VectorMeasure.integral ν (fun x : ℝ => Complex.exp (-((r:ℂ)*(x:ℂ)*Complex.I)))
(ContinuousLinearMap.mul ℝ ℂ)
noncomputable def complexShellEnergy (ν : ComplexMeasure ℝ) (N : ℕ) : ℝ :=
∫ r in PlanarFalconer.dyadicShell N, ‖complexFourier ν r‖^2
noncomputable def complexVariationError (μ : FiniteMeasure ℝ) (ν : ComplexMeasure ℝ) : ℝ :=
(complexOfPositive μ - ν).variation.real univ
end CurrentFalconer.MonotoneShell
end
end
end OAI
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