`BookProof.ChapterShannonSampling.integrableOn_spectrum` (F : Lp ℂ 2 (haarAddCircle (T := T))) : IntegrableOn (fun ξ : ℝ => (F : AddCircle T → ℂ) ξ) (Ioc (-(T / 2)) (-(T / 2) + T))

Proved

BookProof.ChapterShannonSampling.integrableOn_spectrum

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShannonSampling.

BookProof.ChapterShannonSampling.integrableOn_spectrum (F : Lp ℂ 2 (haarAddCircle (T := T))) : IntegrableOn (fun ξ : ℝ => (F : AddCircle T → ℂ) ξ) (Ioc (-(T / 2)) (-(T / 2) + T)) volume

Formalization note: Lean 4 identifier BookProof.ChapterShannonSampling.integrableOn_spectrum.

Preamble

-- Generated from ChapterShannonSampling.lean — theorem BookProof.ChapterShannonSampling.integrableOn_spectrum
import Mathlib
import Definitions.Def_ChapterShannonSampling
open BookProof.ChapterShannonSampling



open MeasureTheory Complex AddCircle intervalIntegral Set
open scoped Real ComplexConjugate

variable {T : ℝ} [hT : Fact (0 < T)]

Formal statement

theorem BookProof.ChapterShannonSampling.integrableOn_spectrum (F : Lp ℂ 2 (haarAddCircle (T := T))) :
    IntegrableOn (fun ξ : ℝ => (F : AddCircle T → ℂ) ξ)
      (Ioc (-(T / 2)) (-(T / 2) + T)) volume := by sorry

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