Inverse of Infinite Hankel Moment Matrices
Let (sₙ)ₙ≥₀ denote an indeterminate Hamburger moment sequence and let H={sₘ₊ₙ} be the corresponding positive definite Hankel matrix. We consider the question of whether there exists an infinite symmetric matrix A = {aⱼ‚ₖ}, which is an inverse of H in the sense that the matrix product AH is defined b...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2018 |
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| Language: | English |
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Інститут математики НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209848 |
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| Cite this: | Inverse of Infinite Hankel Moment Matrices / C. Berg, R. Szwarc // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 25 назв. — англ. |
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Berg, C. Szwarc, R. 2025-11-27T14:52:54Z 2018 Inverse of Infinite Hankel Moment Matrices / C. Berg, R. Szwarc // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 42C05; 44A60; 47B36; 33D45; 60J80 arXiv: 1801.06013 https://nasplib.isofts.kiev.ua/handle/123456789/209848 https://doi.org/10.3842/SIGMA.2018.109 Let (sₙ)ₙ≥₀ denote an indeterminate Hamburger moment sequence and let H={sₘ₊ₙ} be the corresponding positive definite Hankel matrix. We consider the question of whether there exists an infinite symmetric matrix A = {aⱼ‚ₖ}, which is an inverse of H in the sense that the matrix product AH is defined by an absolutely convergent series and AH equals the identity matrix I, a property called (aci). A candidate for A is the coefficient matrix of the reproducing kernel of the moment problem, considered as an entire function of two complex variables. We say that the moment problem has property (aci) if (aci) holds for this matrix A. We show that this is true for many classical indeterminate moment problems but not for the symmetrized version of a cubic birth-and-death process studied by Valent and co-authors. We consider mainly symmetric indeterminate moment problems and give a number of sufficient conditions for (aci) to hold in terms of the recurrence coefficients for the orthonormal polynomials. A sufficient condition is a rapid increase of the recurrence coefficients in the sense that the quotient between consecutive terms is uniformly bounded by a constant strictly smaller than one. We also give a simple example, where (aci) holds, but the inverse matrix of H is highly non-unique. We are grateful to all the referees. Their remarks improved the exposition substantially. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Inverse of Infinite Hankel Moment Matrices Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Inverse of Infinite Hankel Moment Matrices |
| spellingShingle |
Inverse of Infinite Hankel Moment Matrices Berg, C. Szwarc, R. |
| title_short |
Inverse of Infinite Hankel Moment Matrices |
| title_full |
Inverse of Infinite Hankel Moment Matrices |
| title_fullStr |
Inverse of Infinite Hankel Moment Matrices |
| title_full_unstemmed |
Inverse of Infinite Hankel Moment Matrices |
| title_sort |
inverse of infinite hankel moment matrices |
| author |
Berg, C. Szwarc, R. |
| author_facet |
Berg, C. Szwarc, R. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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Article |
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Let (sₙ)ₙ≥₀ denote an indeterminate Hamburger moment sequence and let H={sₘ₊ₙ} be the corresponding positive definite Hankel matrix. We consider the question of whether there exists an infinite symmetric matrix A = {aⱼ‚ₖ}, which is an inverse of H in the sense that the matrix product AH is defined by an absolutely convergent series and AH equals the identity matrix I, a property called (aci). A candidate for A is the coefficient matrix of the reproducing kernel of the moment problem, considered as an entire function of two complex variables. We say that the moment problem has property (aci) if (aci) holds for this matrix A. We show that this is true for many classical indeterminate moment problems but not for the symmetrized version of a cubic birth-and-death process studied by Valent and co-authors. We consider mainly symmetric indeterminate moment problems and give a number of sufficient conditions for (aci) to hold in terms of the recurrence coefficients for the orthonormal polynomials. A sufficient condition is a rapid increase of the recurrence coefficients in the sense that the quotient between consecutive terms is uniformly bounded by a constant strictly smaller than one. We also give a simple example, where (aci) holds, but the inverse matrix of H is highly non-unique.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209848 |
| citation_txt |
Inverse of Infinite Hankel Moment Matrices / C. Berg, R. Szwarc // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 25 назв. — англ. |
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2025-12-07T20:47:49Z |
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2025-12-07T20:47:49Z |
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1850886172441575424 |