`BookProof.ChapterPaleyWienerSampling.bandSignal_congr_ae` {F G : AddCircle T → ℂ} (h : F =ᵐ[haarAddCircle (T := T)] G) (x : ℝ) : bandSignal (T := T) F x = bandSignal (T := T) G x

Proved

BookProof.ChapterPaleyWienerSampling.bandSignal_congr_ae

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterPaleyWienerSampling.

BookProof.ChapterPaleyWienerSampling.bandSignal_congr_ae {F G : AddCircle T → ℂ} (h : F =ᵐ[haarAddCircle (T := T)] G) (x : ℝ) : bandSignal (T := T) F x = bandSignal (T := T) G x

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

Preamble

-- Generated from ChapterPaleyWienerSampling.lean — theorem BookProof.ChapterPaleyWienerSampling.bandSignal_congr_ae
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_congr_ae {F G : AddCircle T → ℂ} (h : F =ᵐ[haarAddCircle (T := T)] G)
    (x : ℝ) : bandSignal (T := T) F x = bandSignal (T := T) G x := by sorry

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