Belton, A and Guillot, D and Khare, A and Putinar, M (2022) Moment-sequence transforms. In: Journal of the European Mathematical Society, 24 (9). pp. 3109-3160.
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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 nonnegative 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 |
|---|---|
| Publication: | Journal of the European Mathematical Society |
| Publisher: | European Mathematical Society Publishing House |
| Additional Information: | The copyright for this article belongs to the Authors. |
| Keywords: | absolutely monotonic function; entrywise function; facewise absolutely monotonic function; Hankel matrix; Laplace transform; moment problem; positive definite matrix; positive polynomial; totally non-negative matrix |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 17 Sep 2022 03:03 |
| Last Modified: | 17 Sep 2022 03:03 |
| URI: | https://eprints.iisc.ac.in/id/eprint/76561 |
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