`BookProof.ChapterShannonSampling.bandSignal_eq_inner` (F : Lp ℂ 2 (haarAddCircle (T := T))) (x : ℝ) : bandSignal (T := T) (F : AddCircle T → ℂ) x = (T : ℂ) * inner ℂ (kernLp (T :=

Proved

BookProof.ChapterShannonSampling.bandSignal_eq_inner

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShannonSampling.

BookProof.ChapterShannonSampling.bandSignal_eq_inner (F : Lp ℂ 2 (haarAddCircle (T := T))) (x : ℝ) : bandSignal (T := T) (F : AddCircle T → ℂ) x = (T : ℂ) * inner ℂ (kernLp (T := T) x) F

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

Preamble

-- Generated from ChapterShannonSampling.lean — theorem BookProof.ChapterShannonSampling.bandSignal_eq_inner
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_inner (F : Lp ℂ 2 (haarAddCircle (T := T))) (x : ℝ) :
    bandSignal (T := T) (F : AddCircle T → ℂ) x
      = (T : ℂ) * inner ℂ (kernLp (T := T) x) F := by sorry

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