Inequalities for exponential polynomials with applications to moment sequences

Abstract

Let $\hat{I}{¦}_{\hat{I}{›}_n}$ be the unique solution of the differential operator $L={\prod }_{j=0}^n\left(\frac{d}{dx}-\hat{I}{»}_j\right)$ such that $\hat{I}{¦}_{\hat{I}{›}_n}^{\left(j\right)}\left(0\right)=0$ for $j=0,...,n-1,$ and $\hat{I}{¦}_{\hat{I}{›}_n}^{\left(n\right)}\left(0\right)=1.$ Assume that $\hat{I}{¦}_{\hat{I}{›}_n}$ is real-valued and $\hat{I}{¦}_{\hat{I}{›}_n}^{\left(n+1\right)}\left(x\right)\ge 0$ for all $x\in \left[0,B\right].$ Then, if a polynomial $R\left(x\right)=\sum _{k=0}^n{a}_k{x}^k$ is non-negative on the interval $\left[0,B\right],$ the inequality \[ {\displaystyle\sum_{k=0}^{n}} a_{k}k!Φ_{Λ_{n}}^{\left( n-k\right) }\left( x\right) \geq R\left( x\right) \] 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 $\hat{I}¼$ over the interval $\left[a,b\right]$ with $b-a<B$ the sequence defined by \[ s_{k}:=\int_{a}^{b}k!Φ_{Λ_{n}}^{\left( n-k\right) }\left( x-a\right) dμ\left( x\right) \] for $k=0,...,n$ is a moment sequence, i.e. there exists a non-negative measure $\hat{I}½$ with support in $\left[a,b\right]$ such that $s_k={\int }_a^b{\left(t-a\right)}^{k}d\hat{I}½\left(t\right)$ for $k=0,....,n.$

14 pages