Inequalities for exponential polynomials with applications to moment sequences

TL;DR

This paper establishes inequalities for exponential polynomials derived from differential operators, with applications to moment sequences and measures, advancing understanding of polynomial bounds and their moment representations.

Contribution

It introduces new inequalities for exponential polynomials based on differential operators and applies these results to characterize moment sequences from measures.

Findings

Derived inequalities for exponential polynomials with differential operator conditions.

Showed that certain sequences from measures are valid moment sequences.

Connected polynomial inequalities to measure support and moment problems.

Abstract

Let $\Phi_{\Lambda_{n}}$ be the unique solution of the differential operator $L=\prod_{j=0}^{n}\left( \frac{d}{dx}-\lambda_{j}\right) $ such that $\Phi_{\Lambda_{n}}^{\left( j\right) }\left( 0\right) =0$ for $j=0,...,n-1,$ and $\Phi_{\Lambda_{n}}^{\left( n\right) }\left( 0\right) =1.$ Assume that $\Phi_{\Lambda_{n}}$ is real-valued and $\Phi_{\Lambda_{n} }^{\left( n+1\right) }\left( x\right) \geq0$ for all $x\in\left[ 0,B\right] .$ Then, if a polynomial $R\left( x\right) = {\displaystyle\sum_{k=0}^{n}} a_{k}x^{k}$ is non-negative on the interval $\left[ 0,B\right] ,$ the inequality \[ {\displaystyle\sum_{k=0}^{n}} a_{k}k!\Phi_{\Lambda_{n}}^{\left( n-k\right) }\left( x\right) \geq R\left( x\right) \] holds for $x\in\left[ 0,B\right] $. From this we derive several interesting inequalities for exponential polynomials. An important consequence is that for a non-negative measure $\mu$ over the interval $\left[ a,b\right] $ with $b-a<B$ the sequence defined by \[ s_{k}:=\int_{a}^{b}k!\Phi_{\Lambda_{n}}^{\left( n-k\right) }\left( x-a\right) d\mu\left( x\right) \] for $k=0,...,n$ is a moment sequence, i.e. there exists a non-negative measure $\nu$ with support in $\left[ a,b\right] $ such that $s_{k}=\int_{a} ^{b}\left( t-a\right) ^{k}d\nu\left( t\right) $ for $k=0,....,n.$