`BookProof.ChapterShannonSampling.inner_kernLp_fourierBasis` (x : ℝ) (n : ℤ) : inner ℂ (kernLp (T := T) x) (fourierBasis (T := T) n) = (sinc (T * x + n) : ℝ)

Proved

BookProof.ChapterShannonSampling.inner_kernLp_fourierBasis

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShannonSampling.

BookProof.ChapterShannonSampling.inner_kernLp_fourierBasis (x : ℝ) (n : ℤ) : inner ℂ (kernLp (T := T) x) (fourierBasis (T := T) n) = (sinc (T * x + n) : ℝ)

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

Preamble

-- Generated from ChapterShannonSampling.lean — theorem BookProof.ChapterShannonSampling.inner_kernLp_fourierBasis
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_fourierBasis (x : ℝ) (n : ℤ) :
    inner ℂ (kernLp (T := T) x) (fourierBasis (T := T) n) = (sinc (T * x + n) : ℝ) := by sorry

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