Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Analysis

Poisson Summation Formula

Periodize a continuous complex-valued function by adding all of its integer translates. If those translates are summable uniformly in sup norm on every compact and the Fourier transform is summable at the integers, then the periodized value at x is reconstructed from those Fourier samples with the phase determined by x modulo one.

Exact theorem

Exact Mathlib statement

theorem Real.tsum_eq_tsum_fourier {f : C(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 (f : ℝ → ℂ) n) (x : ℝ) : ∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 (f : ℝ → ℂ) n * fourier n (x : UnitAddCircle)

The theorem at a glance

Poisson Summation Formula at a glance

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

At every real x, the summable integer-translate series equals the summable Fourier-sample series with its UnitAddCircle phase. Explanatory diagram.
Detailed visual description

The poster states the exact general-x relation in words, keeps continuity and both summability hypotheses prominent, and uses an unlabeled spatial-lattice to period-circle to frequency-lattice diagram. Four source-route movements explain coefficient identification and pointwise Fourier recovery, while the footer reserves the unphased identity for x equals zero.

Statement structure

Statement and scope

Statement map for Poisson Summation FormulaUnder compact-sup-norm summability of integer translates and summability of integer Fourier samples, the translate sum at each real x equals the corresponding phase-weighted Fourier-sample sum. Claim boundary: This page indexes Mathlib's general one-dimensional Poisson summation declaration for a continuous map f : ℝ → ℂ. It assumes that, for every compact K, the series of sup norms of the integer translates restricted to K is summable, and that the series of integer Fourier-transform samples is summable. The conclusion holds at every real x and weights each Fourier sample by fourier n at x viewed in UnitAddCircle. The familiar unphased formula is the x = 0 specialization, not the displayed conclusion at arbitrary x. The pinned upstream declaration is Real.tsum_eq_tsum_fourier. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingUnder compact-sup-normsummability of integertranslates andsummability of integerFourier samples, thetranslate sum at eachreal x equals thecorrespondingphase-weightedFourier-sample sum.Claim boundaryThis page indexesMathlib's generalone-dimensional Poissonsummation declarationfor a continuous map f :ℝ → ℂ. It assumes that,for every compact K, theseries of sup norms ofthe integer translatesrestricted to K issummable, and that theseries of integerFourier-transformsamples is summable. Theconclusion holds atevery real x and weightseach Fourier sample byfourier n at x viewed inUnitAddCircle. Thefamiliar unphasedformula is the x = 0specialization, not thedisplayed conclusion atarbitrary x.Pinned declarationmathlib ·Real.tsum_eq_tsum_fourierStatement map for Poisson Summation FormulaUnder compact-sup-norm summability of integer translates and summability of integer Fourier samples, the translate sum at each real x equals the corresponding phase-weighted Fourier-sample sum. Claim boundary: This page indexes Mathlib's general one-dimensional Poisson summation declaration for a continuous map f : ℝ → ℂ. It assumes that, for every compact K, the series of sup norms of the integer translates restricted to K is summable, and that the series of integer Fourier-transform samples is summable. The conclusion holds at every real x and weights each Fourier sample by fourier n at x viewed in UnitAddCircle. The familiar unphased formula is the x = 0 specialization, not the displayed conclusion at arbitrary x. The pinned upstream declaration is Real.tsum_eq_tsum_fourier. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingUnder compact-sup-normsummability of integertranslates andsummability of integerFourier samples, thetranslate sum at eachreal x equals thecorrespondingphase-weightedFourier-sample sum.Claim boundaryThis page indexesMathlib's generalone-dimensional Poissonsummation declarationfor a continuous map f :ℝ → ℂ. It assumes that,for every compact K, theseries of sup norms ofthe integer translatesrestricted to K issummable, and that theseries of integerFourier-transformsamples is summable. Theconclusion holds atevery real x and weightseach Fourier sample byfourier n at x viewed inUnitAddCircle. Thefamiliar unphasedformula is the x = 0specialization, not thedisplayed conclusion atarbitrary x.Pinned declarationmathlib ·Real.tsum_eq_tsum_fourier

Read the exact Mathlib declaration

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

Integer translates periodize the profile while Fourier samples contribute with phases determined by a general position modulo one. Explanatory scientific diagram.
Detailed visual description

The repeated asymmetric profile represents integer translates of a general continuous function rather than a Gaussian example. The central circle represents periodization modulo one; differently oriented cobalt and emerald spokes represent phase-resolved integer Fourier samples, and the dashed gold spoke marks a generic position. The figure contains no coordinate axes or numeric specialization and does not present the unphased x = 0 identity as the general conclusion.

Why it matters

A mathematical landmark

Poisson summation is a central bridge between spatial and frequency lattices. It converts a sum of translates into a Fourier series and underlies theta-function identities, sampling arguments, analytic number theory, and many duality calculations.

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 page indexes Mathlib's general one-dimensional Poisson summation declaration for a continuous map f : ℝ → ℂ. It assumes that, for every compact K, the series of sup norms of the integer translates restricted to K is summable, and that the series of integer Fourier-transform samples is summable. The conclusion holds at every real x and weights each Fourier sample by fourier n at x viewed in UnitAddCircle. The familiar unphased formula is the x = 0 specialization, not the displayed conclusion at arbitrary x.

Source and local evidence

Where the theorem comes from

Existing declaration
Real.tsum_eq_tsum_fourier in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.poisson-summation-formula.v001
Source
Open the pinned upstream reference