| 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 | |