Inequalities for exponential polynomials with applications to moment sequences
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914446835712000 |
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| author | Kounchev, Ognyan Render, Hermann Tsachev, Tsvetomir |
| author_facet | Kounchev, Ognyan Render, Hermann Tsachev, Tsvetomir |
| contents | 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 $k=0,....,n.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18136 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inequalities for exponential polynomials with applications to moment sequences Kounchev, Ognyan Render, Hermann Tsachev, Tsvetomir Classical Analysis and ODEs Functional Analysis 44A60, 30E05, 42A82 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 $k=0,....,n.$ |
| title | Inequalities for exponential polynomials with applications to moment sequences |
| topic | Classical Analysis and ODEs Functional Analysis 44A60, 30E05, 42A82 |
| url | https://arxiv.org/abs/2409.18136 |