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The strong truncated Hamburger moment problem with and without gaps
ID Zalar, Aljaž (Author)

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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 $\mathbb{R}$, 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 This link opens in a new window
UDC:51
ISSN on article:0022-247X
DOI:10.1016/j.jmaa.2022.126563 This link opens in a new window
COBISS.SI-ID:117448451 This link opens in a new window
Publication date in RUL:19.08.2022
Views:317
Downloads:82
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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 This link opens in a new window

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

Projects

Funder:ARRS - Slovenian Research Agency
Project number:J1-2453
Name:Matrično konveksne množice in realna algebraična geometrija

Funder:ARRS - Slovenian Research Agency
Project number:J1-3004
Name:Hkratna podobnost matrik

Funder:ARRS - Slovenian Research Agency
Project number:P1-0288
Name:Algebre in kolobarji

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