关于指数多项式的不等式及其在矩序列中的应用
Inequalities for exponential polynomials with applications to moment sequences
AI总结：
本文研究了指数多项式不等式，证明了非负多项式在特定区间内满足的不等式，并展示了其在矩序列中的应用。
摘要：
令Φ_{Λ_{n}}是微分算子L=\prod_{j=0}^{n}(\frac{d}{dx}-λ_{j})的唯一解，使得Φ_{Λ_{n}}^{(j)}(0)=0对于j=0,...,n-1，且Φ_{Λ_{n}}^{(n)}(0)=1。假设Φ_{Λ_{n}}是实值的，并且对于所有x∈[0,B]，Φ_{Λ_{n}}^{(n+1)}(x)≥0。然后，如果多项式R(x)=\sum_{k=0}^{n}a_{k}x^{k}在区间[0,B]上非负，则不等式\sum_{k=0}^{n}a_{k}k!Φ_{Λ_{n}}^{(n-k)}(x)≥R(x)在x∈[0,B]上成立。由此我们推导出几个关于指数多项式的有趣不等式。一个重要结论是，对于一个在区间[a,b]上非负的测度μ，其中b-a<B，由s_{k}:=\int_{a}^{b}k!Φ_{Λ_{n}}^{(n-k)}(x-a)dμ(x)定义的序列是矩序列，即存在一个支持在[a,b]上的非负测度ν，使得s_{k}=\int_{a}^{b}(t-a)^{k}dν(t)对于k=0,...,n。
英文摘要：
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 $k=0,....,n.$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
