The strong truncated Hamburger moment problem with and without gaps

Zalar, Aljaž (avtor)

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].

• Jezik: Angleški jezik
• Ključne besede: strong Hamburger moment problem, truncated moment problems, representing measure, moment matrix
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FRI - Fakulteta za računalništvo in informatiko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2022
• Št. strani: 21 str.
• Številčenje: Vol. 516, iss. 2, art. 126563
• PID: 20.500.12556/RUL-138812
• UDK: 51
• ISSN pri članku: 0022-247X
• DOI: 10.1016/j.jmaa.2022.126563
• COBISS.SI-ID: 117448451
• Datum objave v RUL: 19.08.2022

Gradivo je del revije

• Naslov: Journal of mathematical analysis and applications
• Skrajšan naslov: J. math. anal. appl.
• Založnik: Elsevier
• ISSN: 0022-247X
• COBISS.SI-ID: 3081231

Licence

• Licenca: CC BY-NC-ND 4.0, Creative Commons Priznanje avtorstva-Nekomercialno-Brez predelav 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by-nc-nd/4.0/deed.sl
• Opis: Najbolj omejujoča licenca Creative Commons. Uporabniki lahko prenesejo in delijo delo v nekomercialne namene in ga ne smejo uporabiti za nobene druge namene.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: krepki Hamburgerjev momentni problem, prirezani momentni problemi, reprezentirajoča mera, momentna matrika