`BookProof.ChapterSphericalPlancherel.sine_plancherel` (G : 𝓢(ℝ, ℂ)) (hodd : ∀ x, G (-x) = -G x) : ∫ w in Ioi (0 : ℝ), ‖sineTransform (⇑G) w‖ ^ 2 = (1 / 4) * ∫ x in Ioi (0...

Proved

BookProof.ChapterSphericalPlancherel.sine_plancherel

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSphericalPlancherel.

BookProof.ChapterSphericalPlancherel.sine_plancherel (G : 𝓢(ℝ, ℂ)) (hodd : ∀ x, G (-x) = -G x) : ∫ w in Ioi (0 : ℝ), ‖sineTransform (⇑G) w‖ ^ 2 = (1 / 4) * ∫ x in Ioi (0 : ℝ), ‖G x‖ ^ 2

Formalization note: Lean 4 identifier BookProof.ChapterSphericalPlancherel.sine_plancherel.

Preamble

-- Generated from ChapterSphericalPlancherel.lean — theorem BookProof.ChapterSphericalPlancherel.sine_plancherel
import Definitions.Def_ChapterSphericalBessel
import Mathlib
import Definitions.Def_ChapterSphericalPlancherel
open BookProof.ChapterSphericalPlancherel



open MeasureTheory Set Real SchwartzMap
open scoped FourierTransform ContDiff
open BookProof.ChapterSphericalBessel

Formal statement

theorem BookProof.ChapterSphericalPlancherel.sine_plancherel (G : 𝓢(ℝ, ℂ)) (hodd : ∀ x, G (-x) = -G x) :
    ∫ w in Ioi (0 : ℝ), ‖sineTransform (⇑G) w‖ ^ 2
      = (1 / 4) * ∫ x in Ioi (0 : ℝ), ‖G x‖ ^ 2 := by sorry

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