{
  "artifactId": "artifact.library.mathlib.poisson-summation-formula.v001",
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  "category": "Analysis",
  "claimBoundary": "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.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "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)",
  "family": "Fourier analysis",
  "id": "library.mathlib.poisson-summation-formula.v001",
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    "evidence": "/proofs/artifact.library.mathlib.poisson-summation-formula.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.poisson-summation-formula.v001.evidence.json",
    "source": "/sources/upstream/poisson-summation-formula/"
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  "landmark": {
    "completeFormalType": "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)",
    "concepts": [
      "Poisson summation",
      "Fourier transform",
      "periodization",
      "integer lattices",
      "Fourier series",
      "compact-uniform summability"
    ],
    "plainLanguage": "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.",
    "poster": {
      "alt": "An ivory Fourier-analysis poster states the phase-weighted Poisson identity for every real x, lists the two summability hypotheses, and diagrams spatial periodization into a phase-resolved frequency lattice.",
      "byteSize": 1880664,
      "caption": "At every real x, the summable integer-translate series equals the summable Fourier-sample series with its UnitAddCircle phase.",
      "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.",
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      "generatedAt": "2026-07-25T20:37:15Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/poisson-summation-formula/theorem-poster-v1.png",
      "sha256": "sha256:2d6bcc05da3c2d2933d07f7303745ac0ee7457b603615d5db454d0d254942781",
      "title": "Poisson Summation Formula at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "FOURIER ANALYSIS · PERIODIZATION\nPOISSON SUMMATION FORMULA\nf is continuous from the real line to the complex numbers\nInteger translates summable in compact sup norm\nFourier samples summable\nFOR EVERY REAL x\nSUM OVER ALL INTEGERS n OF f(x+n)\n=\nSUM OVER ALL INTEGERS n OF\n[FOURIER TRANSFORM OF f AT n] × [fourier n AT x mod 1]\nHOW THE SOURCE ROUTE MOVES\n1 · Periodize f by summing integer translates\n2 · Compute the Fourier coefficients\n3 · Identify coefficient n with the Fourier transform of f at n\n4 · Recover the Fourier series pointwise\nEXACT SCOPE\nThe right side includes the UnitAddCircle phase.\nThe unphased identity is only x = 0."
    },
    "visual": {
      "alt": "Integer-offset samples of a continuous spatial curve wrap onto a marked period circle and unfold into phase-resolved frequency contributions that combine to the same result.",
      "byteSize": 2024815,
      "caption": "Periodization turns the translate sum into a Fourier series whose coefficients are the integer samples of the Fourier transform.",
      "description": "A continuous curve crosses a disciplined integer lattice inside one compact observation window. Its translates wrap coherently onto a period circle with a marked general position, then fan into frequency contributions with visibly different phases before recombining. A small secondary base-point motif shows phase alignment only as the x equals zero specialization.",
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          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/poisson-summation-formula/theorem-schematic-v1-640.webp",
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          "width": 640
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          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/poisson-summation-formula/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:e8983507593f2f18e1d20df8bd0c24007c44b96cf4a3e1a496dc9c43eabb7a61",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-25T20:37:15Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/poisson-summation-formula/theorem-schematic-v1.png",
      "sha256": "sha256:c6c5d9f5217ae5eb281b7e1a1878924a19ffd465dd5f84e8dd52eddfb0f69c2a",
      "title": "Spatial and frequency lattices through periodization",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "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."
  },
  "nonClaims": [
    "Proof Atlas did not originate the Poisson summation formula or Mathlib's declaration.",
    "The selected declaration does not remove either summability hypothesis or assert compact support, Schwartz decay, or a convergence rate.",
    "It is not a distributional, L², higher-dimensional lattice, or finite discrete Fourier transform statement.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.poisson-summation-formula.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "Under 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.",
  "targetId": "target.library.mathlib.poisson-summation-formula.v001",
  "title": "Poisson Summation Formula",
  "upstreamOrigin": {
    "declarationName": "Real.tsum_eq_tsum_fourier",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/Fourier/PoissonSummation.lean#L102",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
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    "sourceFile": "Mathlib/Analysis/Fourier/PoissonSummation.lean",
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    "status": "reviewed"
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}