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  • 1610.05740

    Rights statement: © 2021 EMS Publishing House. All rights reserved

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    Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License

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Moment-sequence transforms

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Moment-sequence transforms. / Belton, Alexander; Guillot, Dominique; Khare, Apoorva et al.
In: Journal of the European Mathematical Society, Vol. 24, No. 9, 31.05.2022, p. 3109–3160.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Belton, A, Guillot, D, Khare, A & Putinar, M 2022, 'Moment-sequence transforms', Journal of the European Mathematical Society, vol. 24, no. 9, pp. 3109–3160. https://doi.org/10.4171/JEMS/1145

APA

Belton, A., Guillot, D., Khare, A., & Putinar, M. (2022). Moment-sequence transforms. Journal of the European Mathematical Society, 24(9), 3109–3160. https://doi.org/10.4171/JEMS/1145

Vancouver

Belton A, Guillot D, Khare A, Putinar M. Moment-sequence transforms. Journal of the European Mathematical Society. 2022 May 31;24(9):3109–3160. Epub 2021 Sept 8. doi: 10.4171/JEMS/1145

Author

Belton, Alexander ; Guillot, Dominique ; Khare, Apoorva et al. / Moment-sequence transforms. In: Journal of the European Mathematical Society. 2022 ; Vol. 24, No. 9. pp. 3109–3160.

Bibtex

@article{e6bd0752e18b44678714e4e0c8ad1d0b,
title = "Moment-sequence transforms",
abstract = "We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.",
keywords = "Hankel matrix, moment problem, positive definite matrix, totally non-negative matrix, entrywise function, absolutely monotonic function, Laplace transform, positive polynomial, facewise absolutely monotonic function",
author = "Alexander Belton and Dominique Guillot and Apoorva Khare and Mihai Putinar",
note = "{\textcopyright} 2021 EMS Publishing House. All rights reserved",
year = "2022",
month = may,
day = "31",
doi = "10.4171/JEMS/1145",
language = "English",
volume = "24",
pages = "3109–3160",
journal = "Journal of the European Mathematical Society",
issn = "1435-9855",
publisher = "European Mathematical Society Publishing House",
number = "9",

}

RIS

TY - JOUR

T1 - Moment-sequence transforms

AU - Belton, Alexander

AU - Guillot, Dominique

AU - Khare, Apoorva

AU - Putinar, Mihai

N1 - © 2021 EMS Publishing House. All rights reserved

PY - 2022/5/31

Y1 - 2022/5/31

N2 - We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.

AB - We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.

KW - Hankel matrix

KW - moment problem

KW - positive definite matrix

KW - totally non-negative matrix

KW - entrywise function

KW - absolutely monotonic function

KW - Laplace transform

KW - positive polynomial

KW - facewise absolutely monotonic function

U2 - 10.4171/JEMS/1145

DO - 10.4171/JEMS/1145

M3 - Journal article

VL - 24

SP - 3109

EP - 3160

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 9

ER -