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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.07324 |
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| _version_ | 1866916478070030336 |
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| author | Montes-Rodríguez, A. Virtanen, J. A. |
| author_facet | Montes-Rodríguez, A. Virtanen, J. A. |
| contents | We give very simple proofs of the classical results of Magnus and Hill on the spectral properties of the Hilbert matrix $$ H = \left ( {1 \over i+j+ 1 } \right )_{i,j\geq 0} $$ which defines a bounded linear operator on the sequence space $\ell^2$. In particular, we use the Mehler-Fock transform to find the spectrum and the latent eigenfunctions of the Hilbert matrix, that is, we show that the spectrum of $H$ is $[0,π]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=μx$ for some complex number $μ$ (Hill's result). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07324 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Hilbert matrix done right Montes-Rodríguez, A. Virtanen, J. A. Functional Analysis Complex Variables We give very simple proofs of the classical results of Magnus and Hill on the spectral properties of the Hilbert matrix $$ H = \left ( {1 \over i+j+ 1 } \right )_{i,j\geq 0} $$ which defines a bounded linear operator on the sequence space $\ell^2$. In particular, we use the Mehler-Fock transform to find the spectrum and the latent eigenfunctions of the Hilbert matrix, that is, we show that the spectrum of $H$ is $[0,π]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=μx$ for some complex number $μ$ (Hill's result). |
| title | The Hilbert matrix done right |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2411.07324 |