Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Analysis

Fourier Inversion Formula

On a finite-dimensional real inner-product space, transform an integrable function into frequency information and then apply Mathlib's inverse transform. If the transformed function is also integrable, the original value is recovered at each chosen point where the function is continuous.

Exact theorem

Exact Mathlib statement

theorem MeasureTheory.Integrable.fourierInv_fourier_eq {V E : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : V → E} (hf : Integrable f) (h'f : Integrable (𝓕 f)) {v : V} (hv : ContinuousAt f v) : 𝓕⁻ (𝓕 f) v = f v

The theorem at a glance

Fourier inversion at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

Mathlib's inverse transform reconstructs an integrable function at a continuity point when its transform is integrable too. Explanatory diagram.
Detailed visual description

The poster centers the pointwise equation, surrounds it with spatial and frequency representations, and keeps the transform normalization and analytic hypotheses in a distinct scope band. A Gaussian concentration route shows why regularization permits inversion without implying recovery at discontinuities.

Statement structure

Statement and scope

Statement map for Fourier Inversion FormulaIf a function and its Fourier transform are integrable, inverse transformation recovers the function at every point where it is continuous. Claim boundary: This target indexes the pointwise inversion endpoint, not Mathlib's adjacent global continuous-function theorem. It uses volume on a finite-dimensional real inner-product space, requires integrability of f and 𝓕f, completeness of the complex normed codomain, and continuity at the selected point v. The normalization is Mathlib's 2π character convention. The pinned upstream declaration is MeasureTheory.Integrable.fourierInv_fourier_eq. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingIf a function and itsFourier transform areintegrable, inversetransformation recoversthe function at everypoint where it iscontinuous.Claim boundaryThis target indexes thepointwise inversionendpoint, not Mathlib'sadjacent globalcontinuous-functiontheorem. It uses volumeon a finite-dimensionalreal inner-productspace, requiresintegrability of f and𝓕f, completeness of thecomplex normed codomain,and continuity at theselected point v. Thenormalization isMathlib's 2π characterconvention.Pinned declarationmathlib ·MeasureTheory.Integrable.fourierInv_fourier_eqStatement map for Fourier Inversion FormulaIf a function and its Fourier transform are integrable, inverse transformation recovers the function at every point where it is continuous. Claim boundary: This target indexes the pointwise inversion endpoint, not Mathlib's adjacent global continuous-function theorem. It uses volume on a finite-dimensional real inner-product space, requires integrability of f and 𝓕f, completeness of the complex normed codomain, and continuity at the selected point v. The normalization is Mathlib's 2π character convention. The pinned upstream declaration is MeasureTheory.Integrable.fourierInv_fourier_eq. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingIf a function and itsFourier transform areintegrable, inversetransformation recoversthe function at everypoint where it iscontinuous.Claim boundaryThis target indexes thepointwise inversionendpoint, not Mathlib'sadjacent globalcontinuous-functiontheorem. It uses volumeon a finite-dimensionalreal inner-productspace, requiresintegrability of f and𝓕f, completeness of thecomplex normed codomain,and continuity at theselected point v. Thenormalization isMathlib's 2π characterconvention.Pinned declarationmathlib ·MeasureTheory.Integrable.fourierInv_fourier_eq

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

With both integrability hypotheses, inverse Fourier transformation recovers the original value at a continuity point. Explanatory diagram.
Detailed visual description

A smooth vector-valued signal on a finite-dimensional domain flows through a restrained frequency constellation and returns to a second signal. Only one point is highlighted and joined by a precise equality cue, preserving the theorem's pointwise continuity boundary.

Why it matters

A mathematical landmark

Fourier inversion is the reconstruction principle at the heart of harmonic analysis: under precise analytic hypotheses, frequency data determines the original function.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes the pointwise inversion endpoint, not Mathlib's adjacent global continuous-function theorem. It uses volume on a finite-dimensional real inner-product space, requires integrability of f and 𝓕f, completeness of the complex normed codomain, and continuity at the selected point v. The normalization is Mathlib's 2π character convention.

Source and local evidence

Where the theorem comes from

Existing declaration
MeasureTheory.Integrable.fourierInv_fourier_eq in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.fourier-inversion-formula.v001
Source
Open the pinned upstream reference