{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-truncated-moment-problem-on-the-union-of","title":"The truncated moment problem on the union of parallel lines","arxiv_id":"2105.00662","date":"2021-05-03","proceeding":null,"authors":["Aljaž Zalar"],"abstract":"In this article we study the bivariate truncated moment problem (TMP) of degree $2k$ on the union of parallel lines. First we present an alternative proof of Fialkow's solution \\cite{Fia15} to the TMP on the union of two parallel lines (TMP--2pl) using the solution of the truncated Hamburger moment problem (THMP). We add a new equivalent solvability condition, which is then used together with the THMP, to solve the TMP on the union of three parallel lines (TMP--3pl), our second main result of the article. Finally, we establish a sufficient condition for the existence of a solution to the TMP on the union of $n$ parallel lines in the pure case, i.e.\\ when the moment matrix $M_k$ is of the highest possible rank, or equivalently the only column relations come from the union of $n$ lines. The condition is based on the feasibility of a certain linear matrix inequality, corresponding to the extension of $M_k$ by adding rows and columns indexed by some monomials of degree $k+1$. The proof is by induction on $n$, where $n\\geq 2$ and for the base of induction $n=2$ we use the solution of the TMP--2pl.","url_abs":"https://arxiv.org/abs/2105.00662v2","url_pdf":"https://arxiv.org/pdf/2105.00662v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-truncated-moment-problem-on-the-union-of","repo_url":"https://github.com/ZalarA/zalara.github.io","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}