`BookProof.ChapterShannonSampling.bandSignal_eq_of_samples_eq` (F G : Lp ℂ 2 (haarAddCircle (T := T))) (h : ∀ n : ℤ, bandSignal (T := T) (F : AddCircle T → ℂ) (n / T) = bandSignal

Proved

BookProof.ChapterShannonSampling.bandSignal_eq_of_samples_eq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShannonSampling.

BookProof.ChapterShannonSampling.bandSignal_eq_of_samples_eq (F G : Lp ℂ 2 (haarAddCircle (T := T))) (h : ∀ n : ℤ, bandSignal (T := T) (F : AddCircle T → ℂ) (n / T) = bandSignal (T := T) (G : AddCircle T → ℂ) (n / T)) (x : ℝ) : bandSignal (T := T) (F : AddCircle T → ℂ) x = bandSignal (T := T) (G : AddCircle T → ℂ) x

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

Preamble

-- Generated from ChapterShannonSampling.lean — theorem BookProof.ChapterShannonSampling.bandSignal_eq_of_samples_eq
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.bandSignal_eq_of_samples_eq (F G : Lp ℂ 2 (haarAddCircle (T := T)))
    (h : ∀ n : ℤ, bandSignal (T := T) (F : AddCircle T → ℂ) (n / T)
      = bandSignal (T := T) (G : AddCircle T → ℂ) (n / T)) (x : ℝ) :
    bandSignal (T := T) (F : AddCircle T → ℂ) x
      = bandSignal (T := T) (G : AddCircle T → ℂ) x := by sorry

View graph