`BookProof.ChapterPaleyWienerSampling.bandSignal_bandSpectrumLp` (hf : BandLimited T f) (x : ℝ) : bandSignal (T := T) ((bandSpectrumLp hf : Lp ℂ 2 (haarAddCircle (T := T))) : AddCi

Proved

BookProof.ChapterPaleyWienerSampling.bandSignal_bandSpectrumLp

by leonardopedro · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterPaleyWienerSampling.

BookProof.ChapterPaleyWienerSampling.bandSignal_bandSpectrumLp (hf : BandLimited T f) (x : ℝ) : bandSignal (T := T) ((bandSpectrumLp hf : Lp ℂ 2 (haarAddCircle (T := T))) : AddCircle T → ℂ) x = bandSignal (T := T) (bandSpectrum T f) x

Formalization note: Lean 4 identifier BookProof.ChapterPaleyWienerSampling.bandSignal_bandSpectrumLp.

Preamble

-- Generated from ChapterPaleyWienerSampling.lean — theorem BookProof.ChapterPaleyWienerSampling.bandSignal_bandSpectrumLp
import Mathlib
import Definitions.Def_ChapterPaleyWienerSampling
import Definitions.Def_ChapterShannonSampling
open BookProof.ChapterShannonSampling
open BookProof.ChapterPaleyWienerSampling



open MeasureTheory Complex AddCircle Set
open scoped Real FourierTransform
open BookProof.ChapterShannonSampling (sinc bandSignal exists_rep half_add_period)

variable {T : ℝ} {f g : ℝ → ℂ}
variable {T : ℝ} [hT : Fact (0 < T)] {f g : ℝ → ℂ}

Formal statement

theorem BookProof.ChapterPaleyWienerSampling.bandSignal_bandSpectrumLp (hf : BandLimited T f) (x : ℝ) :
    bandSignal (T := T) ((bandSpectrumLp hf : Lp ℂ 2 (haarAddCircle (T := T))) :
        AddCircle T → ℂ) x
      = bandSignal (T := T) (bandSpectrum T f) x := by sorry

View graph