Moment-sequence transforms

  • Alexander Belton

    Lancaster University, UK
  • Dominique Guillot

    University of Delaware, Newark, USA
  • Apoorva Khare

    Analysis & Probability Research Group, Bangalore, India; Indian Institute of Science, Bangalore, India
  • Mihai Putinar

    University of California, Santa Barbara, USA; Newcastle University, Newcastle upon Tyne, UK
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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.

Cite this article

Alexander Belton, Dominique Guillot, Apoorva Khare, Mihai Putinar, Moment-sequence transforms. J. Eur. Math. Soc. 24 (2022), no. 9, pp. 3109–3160

DOI 10.4171/JEMS/1145