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Naslov:The truncated moment problem on curves $y = q(x)$ and $yx^\ell = 1$
Avtorji:ID Zalar, Aljaž (Avtor)
Datoteke:.pdf PDF - Predstavitvena datoteka, prenos (3,50 MB)
MD5: 8BF72E1D07E641A1F286D3DD3FD7D711
 
URL URL - Izvorni URL, za dostop obiščite https://www.tandfonline.com/doi/full/10.1080/03081087.2023.2212316
 
Jezik:Angleški jezik
Tipologija:1.01 - Izvirni znanstveni članek
Organizacija:Logo IMFM - Inštitut za matematiko, fiziko in mehaniko
Povzetek:In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form $y = q(x), q(x) \in \mathbb{R} [x], \deg q ≥ 3$ and $yx^\ell = 1, \ell \in \mathbb{N}$ \ $\{1\}$. For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety $y = x^3$. Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety $y = x^3$. Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: 10.1007/s00020-021-02628-6.; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. doi.org/10.1016/j.laa.2022.05.008.; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: 10.1016/j.jmaa.2022. 126563.], and improve Fialkow’s bound to $\deg q − 1$ (resp. $\ell + 1$) for curves $y = q(x)$ (resp. $yx^\ell = 1$). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are $k \deg q$ (resp. $k(\ell+ 1)$) for curves $y = q(x)$ (resp. $yx^\ell = 1$) for even degree sequences, while for odd ones they are $k \deg q − \bigl \lceil \frac{\deg q}{2} \bigr \rceil$ (resp. $k(\ell + 1) − \bigl \lfloor \frac{\ell}{2} \bigr \rfloor + 1$) for curves $y = q(x)$ (resp. $yx^\ell = 1$). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves $y = x^\ell, \ell = 2, 3,$ and add a new solvability condition to the even degree case on the curve $y = x^2$.
Ključne besede:truncated moment problems, K-moment problems, K-representing measure, minimal measure, moment matrix extensions, positivstellensatz, linear matrix inequality
Status publikacije:Objavljeno
Verzija publikacije:Objavljena publikacija
Datum objave:01.01.2024
Leto izida:2024
Št. strani:str. 1922-1966
Številčenje:Vol. 72, no. 12
PID:20.500.12556/DiRROS-19793 Novo okno
UDK:512
ISSN pri članku:0308-1087
DOI:10.1080/03081087.2023.2212316 Novo okno
COBISS.SI-ID:152329475 Novo okno
Datum objave v DiRROS:25.07.2024
Število ogledov:332
Število prenosov:221
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Gradivo je del revije

Naslov:Linear and Multilinear Algebra
Skrajšan naslov:Linear multilinear algebra
Založnik:Taylor & Francis
ISSN:0308-1087
COBISS.SI-ID:25872128 Novo okno

Gradivo je financirano iz projekta

Financer:ARRS - Agencija za raziskovalno dejavnost Republike Slovenije
Številka projekta:J1-2453
Naslov:Matrično konveksne množice in realna algebraična geometrija

Financer:ARRS - Agencija za raziskovalno dejavnost Republike Slovenije
Številka projekta:J1-3004
Naslov:Hkratna podobnost matrik

Financer:ARRS - Agencija za raziskovalno dejavnost Republike Slovenije
Številka projekta:P1-0288
Naslov:Algebra in njena uporaba

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:prirezani momentni problemi, K-momentni problemi, K-reprezentirajoča mera, minimalna mera, razširitve momentne matrike, linearna matrična neenakost


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