{"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-strong-truncated-hamburger-moment-problem","title":"The strong truncated Hamburger moment problem with and without gaps","arxiv_id":"2101.00486","date":"2021-01-02","proceeding":null,"authors":["Aljaž Zalar"],"abstract":"The strong truncated Hamburger moment problem (STHMP) of degree $(-2k_1,2k_2)$ asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on $\\mathbb{R}\\setminus \\{0\\}$, such that $\\beta_i=\\int x^id\\mu\\; (-2k_1\\leq i\\leq 2k_2)$. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we solve the STHMP. Then, using the equivalence with the STHMP of degree $(-2k,2k)$, we obtain the solution of the 2-dimensional truncated moment problem (TMP) of degree $2k$ with variety $xy=1$, first solved by Curto and Fialkow. Our addition to their result is the fact previously known only for $k=2$, that the existence of a measure is equivalent to the existence of a flat extension of the moment matrix. Further on, we solve the STHMP of degree $(-2k_1,2k_2)$ with one missing moment in the sequence, i.e., $\\beta_{-2k_1+1}$ or $\\beta_{2k_2-1}$, which also gives the solution of the TMP with variety $x^2y=1$ as a special case, first studied by Fialkow.","url_abs":"https://arxiv.org/abs/2101.00486v2","url_pdf":"https://arxiv.org/pdf/2101.00486v2.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-strong-truncated-hamburger-moment-problem","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":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}