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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.06468 |
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| _version_ | 1866908696863309824 |
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| author | Katkova, Olga Vishnyakova, Anna |
| author_facet | Katkova, Olga Vishnyakova, Anna |
| contents | A real sequence $(a_k)_{k=0}^\infty$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix $ \left\| a_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (here $a_k=0$ for $k<0$). In this paper, which continues our earlier work \cite{kv}, we investigate the set of real sequences $(b_k)_{k=0}^\infty$ with the property that for every totally positive sequence $(a_k)_{k=0}^\infty,$ the sequense of termwise products $(a_k b_k)_{k=0}^\infty$ is also totally positive. In particular, we show that for every totally positive sequence $(a_k)_{k=0}^\infty$ the sequence $\left(a_k a^{-k (k-1)}\right)_{k=0}^\infty$ is totally positive whenever $a^2\geq 3{.}503.$ We also propose several open problems concerning convolution operators that preserve total positivity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06468 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convolution operators preserving the set of totally positive sequences Katkova, Olga Vishnyakova, Anna Complex Variables Functional Analysis A real sequence $(a_k)_{k=0}^\infty$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix $ \left\| a_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (here $a_k=0$ for $k<0$). In this paper, which continues our earlier work \cite{kv}, we investigate the set of real sequences $(b_k)_{k=0}^\infty$ with the property that for every totally positive sequence $(a_k)_{k=0}^\infty,$ the sequense of termwise products $(a_k b_k)_{k=0}^\infty$ is also totally positive. In particular, we show that for every totally positive sequence $(a_k)_{k=0}^\infty$ the sequence $\left(a_k a^{-k (k-1)}\right)_{k=0}^\infty$ is totally positive whenever $a^2\geq 3{.}503.$ We also propose several open problems concerning convolution operators that preserve total positivity. |
| title | Convolution operators preserving the set of totally positive sequences |
| topic | Complex Variables Functional Analysis |
| url | https://arxiv.org/abs/2512.06468 |