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.
Highlights
We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line
The absolute-monotonicity conclusion was not new, and resonated with earlier work of Bochner [9] and Schoenberg [42] on positive definite functions on homogeneous spaces. This line of thought was continued by Horn in his doctoral dissertation [25]. These works all address the question of characterizing real functions F which have the property that the matrix (F) is positive semidefinite whenever is, possibly with some structure imposed on these matrices
We focus on functions which preserve moment sequences of positive measures on Euclidean space, or, equivalently, in the one-variable case, functions which leave invariant positive semidefinite Hankel kernels
Summary
We collect the basic concepts and notation necessary for accessing the rest of the article. Studying moment sequences of admissible measures having mass s0 < ρ is equivalent to working with Hankel matrices with entries in a bounded interval (−ρ, ρ) This will be our approach in the remainder of the paper. Transformations which leave invariant Fourier transforms of various classes of measures on groups or homogeneous spaces were studied by many authors, including Schoenberg [42], Bochner [9], Helson, Kahane, Katznelson, and Rudin [23, 28] From the latter works, Rudin extracted [38] an analysis of maps which preserve moment sequences for admissible measures on the torus; equivalently, these are functions which, when applied entrywise, leave invariant the cone of positive semidefinite Toeplitz matrices. Our prior work in fixed dimension has amply exploited the symmetry and combinatorial flavor of similar determinants [3]
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