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The Hilbert matrix done right

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A. Montes-Rodríguez, J. A. Virtanen

Functional Analysis · Submitted 11 Nov 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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 $[0,\pi]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=\mu x$ for some complex number $\mu$ (Hill's result).

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arXiv ID2411.07324 (version 1)
Pith IDXOCFXMBK, Pith's permanent record of this paper and its review
CitationA. Montes-Rodr\'iguez, J. A. Virtanen. "The Hilbert matrix done right." arXiv:2411.07324 (2024). https://pith.science/paper/XOCFXMBK
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,528 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.94 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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 $[0,\pi]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=\mu x$ for some complex number $\mu$ (Hill's result).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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 $\ell^2$. contribution 0.98 0.88 0.21 asserted 1.00
s1 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 $[0,\pi]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=\mu x$ for some complex number $\mu$ (Hill's result). result 0.50 0.81 0.78 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

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 $[0,\pi]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=\mu x$ for some complex number $\mu$ (Hill's result).

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