Public Mathlib landmark · existing upstream theorem · read-only
ProofAtlas

Analysis · existing Mathlib declaration

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.

The theorem at a glance

Poisson Summation Formula at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

At every real x, the summable integer-translate series equals the summable Fourier-sample series with its UnitAddCircle phase. Generated explanation only; the exact HTML statement and pinned source are authoritative.
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.

Formal orientation

Statement map

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. The pinned upstream declaration is Real.tsum_eq_tsum_fourier. The exact checked statement is 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).Mathematical readingUnder compact-sup-normsummability of integertranslates andsummability of integerFourier samples, thetranslate sum at eachreal x equals thecorrespondingphase-weightedFourier-sample sum.Pinned declarationmathlib ·Real.tsum_eq_tsum_fourierExact checked formtheoremReal.tsum_eq_tsum_fourier{f : C(ℝ, ℂ)} (h_norm :∀ K : Compacts ℝ,Summable fun n : ℤ =>‖(f.comp <|ContinuousMap.addRightn).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)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. The pinned upstream declaration is Real.tsum_eq_tsum_fourier. The exact checked statement is 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).Mathematical readingUnder compact-sup-normsummability of integertranslates andsummability of integerFourier samples, thetranslate sum at eachreal x equals thecorrespondingphase-weightedFourier-sample sum.Pinned declarationmathlib ·Real.tsum_eq_tsum_fourierExact checked formtheoremReal.tsum_eq_tsum_fourier{f : C(ℝ, ℂ)} (h_norm :∀ K : Compacts ℝ,Summable fun n : ℤ =>‖(f.comp <|ContinuousMap.addRightn).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)

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

Periodization turns the translate sum into a Fourier series whose coefficients are the integer samples of the Fourier transform. Scientific diagram · explanatory, not proof evidence.
Detailed visual description

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.

Complementary intuition

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.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

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)
Selected evidence declaration text
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)

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

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

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.

Provenance

Origin and evidence stay separate

Existing declaration
Real.tsum_eq_tsum_fourier in mathlib
Relationship
The checked artifact is the upstream declaration itself
Preferred checked artifact
artifact.library.mathlib.poisson-summation-formula.v001
Source handling
Verified upstream reference; no mirrored source package