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

Proved

BookProof.ChapterPaleyWienerSampling.BandLimited.eq_bandSignal_lp

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterPaleyWienerSampling.

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

Formalization note: Lean 4 identifier BookProof.ChapterPaleyWienerSampling.BandLimited.eq_bandSignal_lp.

Preamble

-- Generated from ChapterPaleyWienerSampling.lean — theorem BookProof.ChapterPaleyWienerSampling.BandLimited.eq_bandSignal_lp
import Mathlib
import Definitions.Def_ChapterPaleyWienerSampling
import Definitions.Def_ChapterShannonSampling
open BookProof.ChapterShannonSampling
open BookProof.ChapterPaleyWienerSampling
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.BandLimited.eq_bandSignal_lp (hf : BandLimited T f) (x : ℝ) :
    f x = bandSignal (T := T) ((bandSpectrumLp hf : Lp ℂ 2 (haarAddCircle (T := T))) :
      AddCircle T → ℂ) x := by sorry

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