Inequalities for exponential polynomials with applications to moment sequences
Let $Φ_{Λ_{n}}$ be the unique solution of the differential operator $L=\prod_{j=0}^{n}\left( \frac{d}{dx}-λ_{j}\right) $ such that $Φ_{Λ_{n}}^{\left( j\right) }\left( 0\right) =0$ for $j=0,...,n-1,$ and $Φ_{Λ_{n}}^{\left( n\right) }\left( 0\right) =1.$ Assume that $Φ_{Λ_{n}}$ is real-valued and $Φ_{Λ_{n} }^{\left( n+1\right) }\left( x\right) \geq0$ 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 \[ {\displaystyle\sum_{k=0}^{n}} a_{k}k!Φ_{Λ_{n}}^{\left( n-k\right) }\left( x\right) \geq R\left( x\right) \] 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 $μ$ 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 $ν$ with support in $\left[ a,b\right] $ such that $s_{k}=\int_{a} ^{b}\left( t-a\right) ^{k}dν\left( t\right) $ for $k=0,....,n.$