{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R147",
  "content_hash": null,
  "slug": "ch31-claim-interpolation-bound",
  "type": "claim",
  "title": "The fourth-power norm is at most the square root of the kernel row norm",
  "summary": "The multiplier is an l2 contraction and convolution is bounded on l-infinity by the kernel l1 norm.",
  "relevance": "For Sharp fourth-power norm of the cyclic Hilbert transform at order 31, record ch31-claim-interpolation-bound (“The fourth-power norm is at most the square root of the kernel row norm”) records a bound, answer, status fact, or structural consequence. The record states: The multiplier is an l2 contraction and convolution is bounded on l-infinity by the kernel l1 norm.",
  "relevance_source": "recorded",
  "body": "For odd \\(N=2m+1\\), extend the transform to every complex function by sending the zero mode to zero. Its multiplier has absolute value zero or one, hence Parseval gives\n\\[\n\\|H\\|_{2\\to2}=1.\n\\]\nFourier inversion writes it as convolution with\n\\[\nh_N(j)=\\frac{2}{N}\\sum_{k=1}^{m}\\sin\\frac{2\\pi kj}{N}.\n\\]\nThus \\(|Hf(x)|\\leq\\|h_N\\|_1\\|f\\|_\\infty\\). Interpolating the whole-space bounds at exponents two and infinity gives\n\\[\n\\|H\\|_{4\\to4}\\leq\\|h_N\\|_1^{1/2}.\n\\]\nRestriction to real zero-mean functions can only decrease the operator norm. At \\(N=31\\), interval evaluation gives \\(\\|h_{31}\\|_1<2.707\\), producing the upper endpoint in the bracket.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "family",
    "statement": "odd cyclic groups C_N with the sign Fourier multiplier, specialized numerically to N=31",
    "family": "finite cyclic Hilbert transforms at exponent four"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Direct Fourier inversion, Parseval, the convolution inequality, and Riesz-Thorin interpolation"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Direct Fourier inversion, Parseval, the convolution inequality, and Riesz-Thorin interpolation"
  },
  "relations": [
    {
      "slug": "R146",
      "title": "The sharp fourth-power norm lies between 1.5693 and 1.6453",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "cyclic-hilbert-l4-norm-31",
      "title": "cyclic hilbert l4 norm 31",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}