The strong truncated Hamburger moment problem with and without gaps

Zalar, Aljaž (author)

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 $\mathbb{R}$, such that β$_i$ = ∫ x$^i$dμ (−2k$_1$ ≤ i ≤ 2k$_2$). The first solution of the STHMP, covering also its matrix generalization, was established by Simonov [60], who used the operator approach and described all solutions in terms of self-adjoint extensions of a certain symmetric operator. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we give an alternative solution of the STHMP and describe concretely all minimal solutions, i.e., solutions having the smallest support. 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 [22]. 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., β$_{−2k_1+1}$ or β$_{2k_2−1}$, which also gives the solution of the TMP with variety x$^2$y = 1 as a special case, first studied by Fialkow in [33].

• Language: English
• Keywords: strong Hamburger moment problem, truncated moment problems, representing measure, moment matrix
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FRI - Faculty of Computer and Information Science
• Publication status: Published
• Publication version: Version of Record
• Year: 2022
• Number of pages: 21 str.
• Numbering: Vol. 516, iss. 2, art. 126563
• PID: 20.500.12556/RUL-138812
• UDC: 51
• ISSN on article: 0022-247X
• DOI: 10.1016/j.jmaa.2022.126563
• COBISS.SI-ID: 117448451
• Publication date in RUL: 19.08.2022

Record is a part of a journal

• Title: Journal of mathematical analysis and applications
• Shortened title: J. math. anal. appl.
• Publisher: Elsevier
• ISSN: 0022-247X
• COBISS.SI-ID: 3081231

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: krepki Hamburgerjev momentni problem, prirezani momentni problemi, reprezentirajoča mera, momentna matrika