Pith Number

pith:OA2CIUJG

pith:2026:OA2CIUJG75EWRR57KIOCXRWXVE

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Fourier-based potential theory without an explicit Green's function

Fredrik Fryklund

Potential theory for elliptic PDEs can be formulated from the Fourier symbol alone by parabolic regularization that splits solutions into smooth nonlocal and localized parts.

arxiv:2604.11436 v2 · 2026-04-13 · math.AP · cs.NA · math.NA

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual.

C2weakest assumption

The parabolic regularization produces a valid decomposition into nonlocal and localized components whose asymptotic expansions remain accurate for small ε, under the assumption that the operator belongs to the class of strongly elliptic systems.

C3one line summary

A Fourier-symbol-based potential theory with parabolic regularization decomposes solutions and provides asymptotic expansions for volume, single-layer, and double-layer potentials without explicit Green's functions.

Receipt and verification

First computed	2026-06-23T03:14:28.741808Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7034245126ff4968c7bf521c2bc6d7a901036785e169db2cd68af7585c8ac8b7

Aliases

arxiv: 2604.11436 · arxiv_version: 2604.11436v2 · doi: 10.48550/arxiv.2604.11436 · pith_short_12: OA2CIUJG75EW · pith_short_16: OA2CIUJG75EWRR57 · pith_short_8: OA2CIUJG

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/OA2CIUJG75EWRR57KIOCXRWXVE \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 7034245126ff4968c7bf521c2bc6d7a901036785e169db2cd68af7585c8ac8b7

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6d8584d4c6a63c80290bef354dba7eabe7f26d1f1991ae5875ec7f84cfd77808",
    "cross_cats_sorted": [
      "cs.NA",
      "math.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-04-13T13:20:09Z",
    "title_canon_sha256": "780e6461e00d1922438e4dbae43c280d08a84f0113f27524adbce5ff196bae2b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.11436",
    "kind": "arxiv",
    "version": 2
  }
}