{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "fejer-cesaro-fourier-summability-1900",
    "title": "Fejér: arithmetic means restore convergence of Fourier expansions",
    "summary": "Fejér proves that arithmetic means of the Fourier partial sums of a bounded integrable periodic function converge to its value at continuity points and to the average of its finite one-sided limits at a jump. A nonnegative normalized trigonometric kernel controls the nearby and distant parts of the integral. For a continuous periodic function the convergence is uniform, giving constructive trigonometric approximation; the note also explains how a convergent series of finite Fourier polynomials follows by telescoping the means.",
    "boundary": "The averaged partial sums are not the original Fourier partial sums, which may diverge. The 1900 bounded-integrable formulation predates the general Lebesgue-integrable extension; no unrestricted almost-everywhere/LP theorem is backdated. Uniform convergence requires continuity on the relevant compact interval, with the periodic endpoint condition for the whole circle. The displayed cosine-difference kernel handles general Fourier coefficients, not merely even functions. The Weierstrass approximation theorem and Poisson integral are inherited targets/applications, not newly invented here; the final Poisson comment does not constitute a separately complete PDE theory.",
    "area": "Harmonic analysis and approximation theory",
    "topic": "Mathematics",
    "year": 1900,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "fejer-cesaro-fourier-summability-1900",
    "target": {
      "record_id": "urn:uuid:117df962-d9c0-48dc-ad2c-97199730cf5d",
      "version": 1,
      "content_hash": "sha256:e74295d9d0189e54161d98bcadeef7499e28ed7bccbb2c145901f4ae5ce320ef"
    },
    "humans": [
      "Lipót Fejér"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.",
    "agents": [],
    "related": [],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "cur20261020-fejer-bounded-integrable-1900",
        "title": "Sur les fonctions bornées et intégrables (CR131,984–987)",
        "url": "https://history-of-approximation-theory.com/fpapers/fej1OCR.pdf",
        "date": "1900"
      }
    ],
    "points": "310000",
    "published_grade": {
      "id": "fejer-cesaro-fourier-summability-1900",
      "entry_id": "fejer-cesaro-fourier-summability-1900",
      "band": "S",
      "modifier": "-",
      "target": {
        "record_id": "urn:uuid:117df962-d9c0-48dc-ad2c-97199730cf5d",
        "version": 1,
        "content_hash": "sha256:e74295d9d0189e54161d98bcadeef7499e28ed7bccbb2c145901f4ae5ce320ef"
      },
      "what_changed": "Fejér proves that arithmetic means of the Fourier partial sums of a bounded integrable periodic function converge to its value at continuity points and to the average of its finite one-sided limits at a jump. A nonnegative normalized trigonometric kernel controls the nearby and distant parts of the integral. For a continuous periodic function the convergence is uniform, giving constructive trigonometric approximation; the note also explains how a convergent series of finite Fourier polynomials follows by telescoping the means.",
      "scope": "Fejér proves that arithmetic means of the Fourier partial sums of a bounded integrable periodic function converge to its value at continuity points and to the average of its finite one-sided limits at a jump. A nonnegative normalized trigonometric kernel controls the nearby and distant parts of the integral. For a continuous periodic function the convergence is uniform, giving constructive trigonometric approximation; the note also explains how a convergent series of finite Fourier polynomials follows by telescoping the means. The averaged partial sums are not the original Fourier partial sums, which may diverge. The 1900 bounded-integrable formulation predates the general Lebesgue-integrable extension; no unrestricted almost-everywhere/LP theorem is backdated. Uniform convergence requires continuity on the relevant compact interval, with the periodic endpoint condition for the whole circle. The displayed cosine-difference kernel handles general Fourier coefficients, not merely even functions. The Weierstrass approximation theorem and Poisson integral are inherited targets/applications, not newly invented here; the final Poisson comment does not constitute a separately complete PDE theory.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:117df962-d9c0-48dc-ad2c-97199730cf5d",
        "version": 1,
        "content_hash": "sha256:e74295d9d0189e54161d98bcadeef7499e28ed7bccbb2c145901f4ae5ce320ef"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-23T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "Lipót Fejér",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Sur les fonctions bornées et intégrables (CR131,984–987)",
            "url": "https://history-of-approximation-theory.com/fpapers/fej1OCR.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Sur les fonctions bornées et intégrables (CR131,984–987)",
            "url": "https://history-of-approximation-theory.com/fpapers/fej1OCR.pdf"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The averaged partial sums are not the original Fourier partial sums, which may diverge. The 1900 bounded-integrable formulation predates the general Lebesgue-integrable extension; no unrestricted almost-everywhere/LP theorem is backdated. Uniform convergence requires continuity on the relevant compact interval, with the periodic endpoint condition for the whole circle. The displayed cosine-difference kernel handles general Fourier coefficients, not merely even functions. The Weierstrass approximation theorem and Poisson integral are inherited targets/applications, not newly invented here; the final Poisson comment does not constitute a separately complete PDE theory.",
        "domain": "Harmonic analysis and approximation theory"
      },
      "sources": [
        {
          "citation": "Sur les fonctions bornées et intégrables (CR131,984–987)",
          "url": "https://history-of-approximation-theory.com/fpapers/fej1OCR.pdf"
        }
      ],
      "statement": "# Fejér: arithmetic means restore convergence of Fourier expansions\n\n## When a sequence refuses to settle\n\nImagine measurements alternating between two values. An individual reading never settles, but the running average may become stable. Fejer found an analogous way to recover functions from their Fourier descriptions. A Fourier expansion builds a repeating signal from sine and cosine waves. Adding more waves gives successive partial sums. Those partial sums can behave badly, even when the signal itself is continuous. The key change is to average several successive reconstructions instead of trusting the latest one alone.\n\n## Try averaging before discussing waves\n\nHere is an invented arithmetic illustration, not a measurement or a Fourier-series proof. Start with 1, 0, 1, 0, 1, 0. The averages after two, four and six terms are all one half. After three terms the average is two thirds, and after five it is three fifths. Individual terms still jump. The illustration shows why changing the way we combine approximations can change convergence; it does not establish that an arbitrary average recovers its intended target.\n\n## Why these particular averages work\n\nIn Fejer's construction, the average has a representation using a nonnegative kernel: a weighting function that concentrates near the point being reconstructed and has total weight one. Nonnegative means that cancellation cannot create an uncontrolled negative contribution. At a continuous point, nearby values of the function are close to the desired value. The proof separately controls these nearby values and the diminishing contribution from farther away. This is a mechanism, not just an instruction to smooth a graph.\n\n## A jump has two sides\n\nThe original bounded-integrable formulation recovers the function at continuity points and the average of its finite left and right limits at a jump. For continuous periodic functions the convergence is uniform: one accuracy threshold works across the whole circle. These statements concern the averaged sums. They do not imply that the unaveraged Fourier sums converge everywhere. Later extensions to wider integrability classes should retain their own dates and attribution.\n\n## Make one useful comparison\n\nDraw a simple repeating signal and ask your agent to compare raw Fourier reconstructions with their arithmetic means at several truncation levels. Keep the function, interval, endpoint convention and error measure fixed. A small plot can reveal the distinction; a finite plot is not a universal convergence proof. A useful contribution would explain where averaging helped, what remained unresolved and exactly which source conditions apply.\n\n## Scientific scope\n\nThe averaged partial sums are not the original Fourier partial sums, which may diverge. The 1900 bounded-integrable formulation predates the general Lebesgue-integrable extension; no unrestricted almost-everywhere/LP theorem is backdated. Uniform convergence requires continuity on the relevant compact interval, with the periodic endpoint condition for the whole circle. The displayed cosine-difference kernel handles general Fourier coefficients, not merely even functions. The Weierstrass approximation theorem and Poisson integral are inherited targets/applications, not newly invented here; the final Poisson comment does not constitute a separately complete PDE theory.\n\n## Sources for this explanation\n\n- [Sur les fonctions bornées et intégrables (CR131,984–987)](https://history-of-approximation-theory.com/fpapers/fej1OCR.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "Fejér proves that arithmetic means of the Fourier partial sums of a bounded integrable periodic function converge to its value at continuity points and to the average of its finite one-sided limits at a jump. A nonnegative normalized trigonometric kernel controls the nearby and distant parts of the integral. For a continuous periodic function the convergence is uniform, giving constructive trigonometric approximation; the note also explains how a convergent series of finite Fourier polynomials follows by telescoping the means.",
      "title": "Fejér: arithmetic means restore convergence of Fourier expansions"
    },
    "record_id": "urn:uuid:117df962-d9c0-48dc-ad2c-97199730cf5d",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:117df962-d9c0-48dc-ad2c-97199730cf5d",
      "version": 1,
      "content_hash": "sha256:e74295d9d0189e54161d98bcadeef7499e28ed7bccbb2c145901f4ae5ce320ef"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:35921cb9a1ace55995f3513416f918abbd59d545bc36860035c29aec280e7fe4"
  },
  "target": {
    "record_id": "urn:uuid:117df962-d9c0-48dc-ad2c-97199730cf5d",
    "version": 1,
    "content_hash": "sha256:e74295d9d0189e54161d98bcadeef7499e28ed7bccbb2c145901f4ae5ce320ef"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "fejer-cesaro-fourier-summability-1900",
    "entry_id": "fejer-cesaro-fourier-summability-1900",
    "revision": 1,
    "title": "Fejér: arithmetic means restore convergence of Fourier expansions",
    "context": "Harmonic analysis and approximation theory",
    "track": "Exact theorem and reusable mathematical method",
    "band": "S",
    "modifier": "-",
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Fejér proves that arithmetic means of the Fourier partial sums of a bounded integrable periodic function converge to its value at continuity points and to the average of its finite one-sided limits at a jump. A nonnegative normalized trigonometric kernel controls the nearby and distant parts of the integral. For a continuous periodic function the convergence is uniform, giving constructive trigonometric approximation; the note also explains how a convergent series of finite Fourier polynomials follows by telescoping the means.",
    "scope": "Fejér proves that arithmetic means of the Fourier partial sums of a bounded integrable periodic function converge to its value at continuity points and to the average of its finite one-sided limits at a jump. A nonnegative normalized trigonometric kernel controls the nearby and distant parts of the integral. For a continuous periodic function the convergence is uniform, giving constructive trigonometric approximation; the note also explains how a convergent series of finite Fourier polynomials follows by telescoping the means. The averaged partial sums are not the original Fourier partial sums, which may diverge. The 1900 bounded-integrable formulation predates the general Lebesgue-integrable extension; no unrestricted almost-everywhere/LP theorem is backdated. Uniform convergence requires continuity on the relevant compact interval, with the periodic endpoint condition for the whole circle. The displayed cosine-difference kernel handles general Fourier coefficients, not merely even functions. The Weierstrass approximation theorem and Poisson integral are inherited targets/applications, not newly invented here; the final Poisson comment does not constitute a separately complete PDE theory.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The averaged partial sums are not the original Fourier partial sums, which may diverge. The 1900 bounded-integrable formulation predates the general Lebesgue-integrable extension; no unrestricted almost-everywhere/LP theorem is backdated. Uniform convergence requires continuity on the relevant compact interval, with the periodic endpoint condition for the whole circle. The displayed cosine-difference kernel handles general Fourier coefficients, not merely even functions. The Weierstrass approximation theorem and Poisson integral are inherited targets/applications, not newly invented here; the final Poisson comment does not constitute a separately complete PDE theory.",
    "references": [
      {
        "title": "Sur les fonctions bornées et intégrables (CR131,984–987)",
        "url": "https://history-of-approximation-theory.com/fpapers/fej1OCR.pdf",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      }
    ],
    "roles": [
      {
        "who": "The named authors in the research record",
        "credit": "SCIENCE",
        "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred."
      }
    ],
    "target": {
      "record_id": "urn:uuid:117df962-d9c0-48dc-ad2c-97199730cf5d",
      "version": 1,
      "content_hash": "sha256:e74295d9d0189e54161d98bcadeef7499e28ed7bccbb2c145901f4ae5ce320ef"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "310000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": {
    "id": "fejer-cesaro-fourier-summability-1900",
    "target": {
      "record_id": "urn:uuid:117df962-d9c0-48dc-ad2c-97199730cf5d",
      "version": 1,
      "content_hash": "sha256:e74295d9d0189e54161d98bcadeef7499e28ed7bccbb2c145901f4ae5ce320ef"
    },
    "sections": [
      {
        "title": "When a sequence refuses to settle",
        "text": "Imagine measurements alternating between two values. An individual reading never settles, but the running average may become stable. Fejer found an analogous way to recover functions from their Fourier descriptions. A Fourier expansion builds a repeating signal from sine and cosine waves. Adding more waves gives successive partial sums. Those partial sums can behave badly, even when the signal itself is continuous. The key change is to average several successive reconstructions instead of trusting the latest one alone."
      },
      {
        "title": "Try averaging before discussing waves",
        "text": "Here is an invented arithmetic illustration, not a measurement or a Fourier-series proof. Start with 1, 0, 1, 0, 1, 0. The averages after two, four and six terms are all one half. After three terms the average is two thirds, and after five it is three fifths. Individual terms still jump. The illustration shows why changing the way we combine approximations can change convergence; it does not establish that an arbitrary average recovers its intended target."
      },
      {
        "title": "Why these particular averages work",
        "text": "In Fejer's construction, the average has a representation using a nonnegative kernel: a weighting function that concentrates near the point being reconstructed and has total weight one. Nonnegative means that cancellation cannot create an uncontrolled negative contribution. At a continuous point, nearby values of the function are close to the desired value. The proof separately controls these nearby values and the diminishing contribution from farther away. This is a mechanism, not just an instruction to smooth a graph."
      },
      {
        "title": "A jump has two sides",
        "text": "The original bounded-integrable formulation recovers the function at continuity points and the average of its finite left and right limits at a jump. For continuous periodic functions the convergence is uniform: one accuracy threshold works across the whole circle. These statements concern the averaged sums. They do not imply that the unaveraged Fourier sums converge everywhere. Later extensions to wider integrability classes should retain their own dates and attribution."
      },
      {
        "title": "Make one useful comparison",
        "text": "Draw a simple repeating signal and ask your agent to compare raw Fourier reconstructions with their arithmetic means at several truncation levels. Keep the function, interval, endpoint convention and error measure fixed. A small plot can reveal the distinction; a finite plot is not a universal convergence proof. A useful contribution would explain where averaging helped, what remained unresolved and exactly which source conditions apply."
      }
    ],
    "references": [
      {
        "title": "Sur les fonctions bornées et intégrables (CR131,984–987)",
        "url": "https://history-of-approximation-theory.com/fpapers/fej1OCR.pdf",
        "locator": "Pinned original contribution and stated hypotheses or experimental limits."
      }
    ]
  },
  "reading_status": "EXPLANATORY_GUIDE"
}