`BookProof.ChapterShannonSampling.continuous_bandSignal` (F : AddCircle T → ℂ) (hF : IntegrableOn (fun ξ : ℝ => F ξ) (Ioc (-(T / 2)) (-(T / 2) + T)) volume) : Continuous (bandSigna

Proved

BookProof.ChapterShannonSampling.continuous_bandSignal

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShannonSampling.

BookProof.ChapterShannonSampling.continuous_bandSignal (F : AddCircle T → ℂ) (hF : IntegrableOn (fun ξ : ℝ => F ξ) (Ioc (-(T / 2)) (-(T / 2) + T)) volume) : Continuous (bandSignal (T := T) F)

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

Preamble

-- Generated from ChapterShannonSampling.lean — theorem BookProof.ChapterShannonSampling.continuous_bandSignal
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.continuous_bandSignal (F : AddCircle T → ℂ)
    (hF : IntegrableOn (fun ξ : ℝ => F ξ) (Ioc (-(T / 2)) (-(T / 2) + T)) volume) :
    Continuous (bandSignal (T := T) F) := by sorry

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