Hermite-Poulain theorems for linear finite difference operators
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866909663211028480 |
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| author | Katkova, Olga Tyaglov, Mikhail Vishnyakova, Anna |
| author_facet | Katkova, Olga Tyaglov, Mikhail Vishnyakova, Anna |
| contents | We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form $$ Δ_{θ, h}(f)(z)=e^{iθ}f(z+ih)-e^{-iθ}f(z-ih), \quadθ\in[0,π),\ \ h\in\mathbb{C}\setminus\{0\}, $$ where $f$ is a polynomial or an entire function of a certain kind, and prove that the roots of $Δ_{θ, h}(f)$ are simple under some conditions. Moreover, we prove that the operator $Δ_{θ, h}$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of $Δ_{θ, h}(p)$ as $|h|\to\infty$ is found for any complex polynomial $p$. Some other interesting roots preserving properties of the operator $Δ_{θ, h}$ are also studied, and a few examples are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1901_06398 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Hermite-Poulain theorems for linear finite difference operators Katkova, Olga Tyaglov, Mikhail Vishnyakova, Anna Classical Analysis and ODEs 30C15, 30D15, 30D35, 26C10, 16C10 We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form $$ Δ_{θ, h}(f)(z)=e^{iθ}f(z+ih)-e^{-iθ}f(z-ih), \quadθ\in[0,π),\ \ h\in\mathbb{C}\setminus\{0\}, $$ where $f$ is a polynomial or an entire function of a certain kind, and prove that the roots of $Δ_{θ, h}(f)$ are simple under some conditions. Moreover, we prove that the operator $Δ_{θ, h}$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of $Δ_{θ, h}(p)$ as $|h|\to\infty$ is found for any complex polynomial $p$. Some other interesting roots preserving properties of the operator $Δ_{θ, h}$ are also studied, and a few examples are presented. |
| title | Hermite-Poulain theorems for linear finite difference operators |
| topic | Classical Analysis and ODEs 30C15, 30D15, 30D35, 26C10, 16C10 |
| url | https://arxiv.org/abs/1901.06398 |