A counterexample to the weak Shanks conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910460275589120 |
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| author | Bénéteau, Catherine Khavinson, Dmitry Seco, Daniel |
| author_facet | Bénéteau, Catherine Khavinson, Dmitry Seco, Daniel |
| contents | We give an example of a function $f$ non-vanishing in the closed bidisk and the affine polynomial minimizing the norm of $1-pf$ in the Hardy space of the bidisk among all affine polynomials $p$. We show that this polynomial vanishes inside the bidisk. This provides a counterexample to the weakest form of a conjecture due to Shanks that has been open since 1980, with applications that arose from digital filter design. This counterexample has a simple form and follows naturally from [7], where the phenomenon of zeros seeping into the unit disk was already observed for similar minimization problems in one variable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_16943 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A counterexample to the weak Shanks conjecture Bénéteau, Catherine Khavinson, Dmitry Seco, Daniel Complex Variables Classical Analysis and ODEs Functional Analysis Primary 32A35, Secondary 32A60, 47A16, 30C15 We give an example of a function $f$ non-vanishing in the closed bidisk and the affine polynomial minimizing the norm of $1-pf$ in the Hardy space of the bidisk among all affine polynomials $p$. We show that this polynomial vanishes inside the bidisk. This provides a counterexample to the weakest form of a conjecture due to Shanks that has been open since 1980, with applications that arose from digital filter design. This counterexample has a simple form and follows naturally from [7], where the phenomenon of zeros seeping into the unit disk was already observed for similar minimization problems in one variable. |
| title | A counterexample to the weak Shanks conjecture |
| topic | Complex Variables Classical Analysis and ODEs Functional Analysis Primary 32A35, Secondary 32A60, 47A16, 30C15 |
| url | https://arxiv.org/abs/2405.16943 |