Moment-sequence transforms

Belton, Alexander and Guillot, Dominique and Khare, Apoorva and Putinar, Mihai (2022) Moment-sequence transforms. Journal of the European Mathematical Society, 24 (9). 3109–3160. ISSN 1435-9855

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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.

Item Type:
Journal Article
Journal or Publication Title:
Journal of the European Mathematical Society
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© 2021 EMS Publishing House. All rights reserved
Uncontrolled Keywords:
?? hankel matrixmoment problempositive definite matrixtotally non-negative matrixentrywise functionabsolutely monotonic functionlaplace transformpositive polynomialfacewise absolutely monotonic functionmathematics(all)analysisalgebra and number theoryapplied ??
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Deposited On:
10 Mar 2021 15:00
Last Modified:
03 Jul 2024 00:03