The pure $Y=X^{d}$ truncated moment problem
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909952534118400 |
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| author | Fialkow, Lawrence Zalar, Aljaž |
| author_facet | Fialkow, Lawrence Zalar, Aljaž |
| contents | Let $β\equivβ^{(2n)}$ be a real bivariate sequence of degree $2n$. We study the existence of representing measures for $β$ supported in the curve $y=x^{d}$ ($d\ge 1$) in the case when all column dependence relations in the moment matrix $M_n(β)$ are generated by the relation $Y=X^{d}$. We prove that the core variety of $β$, $\mathcal{CV}(L_β)$, is nonempty (equivalently, representing measures exist) if and only if $C$, the partially defined core matrix of $β$, admits a positive, recursively generated completion $C[A]$. Moreover, $\mathcal{CV}(L_β)$ is the entire curve $y=x^{d}$ if and only if there is a positive definite completion $C[A]$. In the remaining case, if there is a measure, it is unique and finitely atomic. For $d = 3$, we use these results to compute the core variety of $β$ and give new characterizations of the existence of representing measures, which complement a result of the first-named author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10375 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The pure $Y=X^{d}$ truncated moment problem Fialkow, Lawrence Zalar, Aljaž Functional Analysis Primary 44A60, 47A57, 47A20, Secondary 47N40 Let $β\equivβ^{(2n)}$ be a real bivariate sequence of degree $2n$. We study the existence of representing measures for $β$ supported in the curve $y=x^{d}$ ($d\ge 1$) in the case when all column dependence relations in the moment matrix $M_n(β)$ are generated by the relation $Y=X^{d}$. We prove that the core variety of $β$, $\mathcal{CV}(L_β)$, is nonempty (equivalently, representing measures exist) if and only if $C$, the partially defined core matrix of $β$, admits a positive, recursively generated completion $C[A]$. Moreover, $\mathcal{CV}(L_β)$ is the entire curve $y=x^{d}$ if and only if there is a positive definite completion $C[A]$. In the remaining case, if there is a measure, it is unique and finitely atomic. For $d = 3$, we use these results to compute the core variety of $β$ and give new characterizations of the existence of representing measures, which complement a result of the first-named author. |
| title | The pure $Y=X^{d}$ truncated moment problem |
| topic | Functional Analysis Primary 44A60, 47A57, 47A20, Secondary 47N40 |
| url | https://arxiv.org/abs/2508.10375 |