`BookProof.ChapterShannonSampling.inner_kernLp_apply` (F : Lp ℂ 2 (haarAddCircle (T := T))) (x : ℝ) : inner ℂ (kernLp (T := T) x) F = ∫ z : AddCircle T, conj (kern (T := T) x z) *

Proved

BookProof.ChapterShannonSampling.inner_kernLp_apply

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShannonSampling.

BookProof.ChapterShannonSampling.inner_kernLp_apply (F : Lp ℂ 2 (haarAddCircle (T := T))) (x : ℝ) : inner ℂ (kernLp (T := T) x) F = ∫ z : AddCircle T, conj (kern (T := T) x z) * F z ∂haarAddCircle

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

Preamble

-- Generated from ChapterShannonSampling.lean — theorem BookProof.ChapterShannonSampling.inner_kernLp_apply
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.inner_kernLp_apply (F : Lp ℂ 2 (haarAddCircle (T := T))) (x : ℝ) :
    inner ℂ (kernLp (T := T) x) F
      = ∫ z : AddCircle T, conj (kern (T := T) x z) * F z ∂haarAddCircle := by sorry

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