The Hadamard Product of Moment Sequences, Diagonal Positivity Preservers, and their Generators

2024/07/29 by Philipp J. di Dio, di Dio, Philipp J., Langer, Lars-Luca · 2 citations
Computer Science · #Algebraic Geometry (math.AG) #Computational Geometry and Mesh Generation #Data Management and Algorithms #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2407.19933

openalex publication_date 2024/07/29 · openalex created_date 2024/08/01 · openalex updated_date 2026/07/28

Abstract

In this work we investigate special aspects of positivity preservers and especially diagonal positivity preservers, i.e., linear maps T:ℝ[x₁,…,x_n]→ℝ[x₁,…,x_n] such that Tx^α= t_αx^α holds for all α∈ℕ₀ⁿ with t_α∈ℝ and Tp≥ 0 on ℝⁿ for all p∈ℝ[x₁,…,x_n] with p≥ 0 on ℝⁿ. We discuss representations of T, give characterizations of diagonal positivity preservers, and compare these to previous (partial) results in the literature. On the side we get a full characterization of linear maps preserving moment sequences and a new proof of Schur's product formula. The tool of diagonal positivity preservers simplifies several other existing proofs in the literature. We give a full characterization of generators A of diagonal positivity preservers, i.e., e^tA is a diagonal positivity preserver for all t≥ 0. We give the connection of these generators to infinitely divisible moment sequences.

Cited by

• A Classification of Elements of Sequence Space Seq(ℝ)
• Making Non-Negative Polynomials into Sums of Squares

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