| import OAI.Analysis.FourierLLogL.Main | |
| open MeasureTheory Filter | |
| open scoped BigOperators Topology | |
| namespace OAI.FourierLLogL | |
| theorem paperMain (f : AddCircle (2 * Real.pi) → ℂ) | |
| (hfm : AEStronglyMeasurable f AddCircle.haarAddCircle) | |
| (hfi : Integrable (fun x ↦ ‖f x‖ * Real.log (2 + ‖f x‖)) | |
| AddCircle.haarAddCircle) : | |
| ∃ E : Set (AddCircle (2 * Real.pi)), | |
| MeasurableSet E ∧ AddCircle.haarAddCircle E = 0 ∧ ∀ x ∉ E, | |
| Tendsto | |
| (fun N : ℕ ↦ | |
| ∑ k ∈ Finset.Icc (-(N : ℤ)) (N : ℤ), fourierCoeff f k * fourier k x) | |
| atTop (𝓝 (f x)) := by | |
| simpa only [circleMeasure, symmetricPartialSum] using | |
| (almost_everywhere_fourier_convergence f ⟨hfm, hfi⟩) | |
| end OAI.FourierLLogL | |