Pith Number

pith:Q7XIP65Y

pith:2022:Q7XIP65YXDZUQA2BY57WFJ7XOO

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Bernstein-Szeg\H{o} measures in the plane

Jeffrey S. Geronimo, Plamen Iliev

Bernstein-Szegő measures on R² are defined via a new identity linking Fejér-Riesz factorization of the weight to a three-variable polynomial, yielding explicit orthonormal bases and complete characterization by finitely many moments.

arxiv:2207.14383 v4 · 2022-07-28 · math.CA · math.CV · math.FA

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Claims

C1strongest claim

We define a class of Bernstein-Szegő measures on R² and establish their spectral properties, providing a natural extension of the one-dimensional theory. We also derive conditions involving finitely many moments, which are new in the two-dimensional setting, and which completely characterize these measures.

C2weakest assumption

That a new identity exists connecting a Fejér-Riesz factorization of the weight to a polynomial in three variables associated with the measure, and that recent bivariate trigonometric Fejér-Riesz results suffice to define a nontrivial two-dimensional Szegő mapping yielding explicit orthonormal bases.

C3one line summary

Defines Bernstein-Szegő measures on R², derives new finite-moment characterization conditions, and constructs orthonormal bases via an extended Szegő mapping from bivariate Fejér-Riesz factorization.

References

28 extracted · 28 resolved · 0 Pith anchors

[1] Ju. M. Berezans ′ki ˘ ı,Expansions in eigenfunctions of selfadjoint operators , Translations of Mathematical Monographs, Vol. 17, American Mathematical S ociety, Providence, R.I., 1968 1968

[2] D. Damanik, A. Pushnitski and B. Simon, The analytic theory of matrix orthogonal polyno- mials, Surv. Approx. Theory 4 (2008), 1–85 2008

[3] D. Damanik and B. Simon, Jost functions and Jost solutions for Jacobi matrices. II. D ecay and analyticity , Int. Math. Res. Not. 2006, Art. ID 19396, 32 pp 2006

[4] A. Delgado, J. Geronimo, P. Iliev and F. Marcell´ an, Two variable orthogonal polynomials and structured matrices , SIAM J. Matr. Anal. Appl. 28 (2006), no. 1, 118–147 2006

[5] A. Delgado, J. Geronimo, P. Iliev and Y. Xu, On a two variable class of Bernstein-Szeg˝ o measures, Constr. Approx. 30 (2009), no. 1, 71–91 2009

Receipt and verification

First computed	2026-06-03T01:05:42.286844Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

87ee87fbb8b8f3480341c77f62a7f77387e50acf1d7202033ccf595b50578e06

Aliases

arxiv: 2207.14383 · arxiv_version: 2207.14383v4 · doi: 10.48550/arxiv.2207.14383 · pith_short_12: Q7XIP65YXDZU · pith_short_16: Q7XIP65YXDZUQA2B · pith_short_8: Q7XIP65Y

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/Q7XIP65YXDZUQA2BY57WFJ7XOO \
  | 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: 87ee87fbb8b8f3480341c77f62a7f77387e50acf1d7202033ccf595b50578e06

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "80c374a2de1185561ceb73db20cee6697a56c0e52d453915dda5a24c54c7e378",
    "cross_cats_sorted": [
      "math.CV",
      "math.FA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CA",
    "submitted_at": "2022-07-28T21:25:52Z",
    "title_canon_sha256": "718878b51724203bf09beae0e02140e6f5d780fffc08022cc8f49279728c967d"
  },
  "schema_version": "1.0",
  "source": {
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    "kind": "arxiv",
    "version": 4
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}