`BookProof.ChapterShannonSampling.bandSignal_eq_circle_integral` (F : AddCircle T → ℂ) (x : ℝ) : bandSignal (T := T) F x = ∫ z : AddCircle T, conj (kern (T := T) x z) * F z

Proved

BookProof.ChapterShannonSampling.bandSignal_eq_circle_integral

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShannonSampling.

BookProof.ChapterShannonSampling.bandSignal_eq_circle_integral (F : AddCircle T → ℂ) (x : ℝ) : bandSignal (T := T) F x = ∫ z : AddCircle T, conj (kern (T := T) x z) * F z

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

Preamble

-- Generated from ChapterShannonSampling.lean — theorem BookProof.ChapterShannonSampling.bandSignal_eq_circle_integral
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_circle_integral (F : AddCircle T → ℂ) (x : ℝ) :
    bandSignal (T := T) F x = ∫ z : AddCircle T, conj (kern (T := T) x z) * F z := by sorry

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