`BookProof.ChapterShannonSampling.bandSignal_sample` (F : AddCircle T → ℂ) (n : ℤ) : bandSignal (T := T) F (-(n / T)) = (T : ℂ) * fourierCoeff F n

Proved

BookProof.ChapterShannonSampling.bandSignal_sample

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShannonSampling.

BookProof.ChapterShannonSampling.bandSignal_sample (F : AddCircle T → ℂ) (n : ℤ) : bandSignal (T := T) F (-(n / T)) = (T : ℂ) * fourierCoeff F n

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

Preamble

-- Generated from ChapterShannonSampling.lean — theorem BookProof.ChapterShannonSampling.bandSignal_sample
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_sample (F : AddCircle T → ℂ) (n : ℤ) :
    bandSignal (T := T) F (-(n / T)) = (T : ℂ) * fourierCoeff F n := by sorry

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