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Concrete solution to the nonsingular quartic binary moment problem


Authors: Raúl E. Curto and Seonguk Yoo
Journal: Proc. Amer. Math. Soc. 144 (2016), 249-258
MSC (2010): Primary 47A57, 44A60, 42A70, 30A05; Secondary 15A15, 15-04, 47N40, 47A20
DOI: https://doi.org/10.1090/proc/12698
Published electronically: June 30, 2015
MathSciNet review: 3415593
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Abstract: Given real numbers $ \beta \equiv \beta ^{\left ( 4\right ) }\colon \beta _{00}$, $ \beta _{10}$, $ \beta _{01}$, $ \beta _{20}$, $ \beta _{11}$, $ \beta _{02}$, $ \beta _{30}$, $ \beta _{21}$, $ \beta _{12}$, $ \beta _{03}$, $ \beta _{40}$, $ \beta _{31}$, $ \beta _{22}$, $ \beta _{13}$, $ \beta _{04}$, with $ \beta _{00} >0$, the quartic real moment problem for $ \beta $ entails finding conditions for the existence of a positive Borel measure $ \mu $, supported in $ \mathbb{R}^2$, such that $ \beta _{ij}=\int s^{i}t^{j}\,d\mu \;\;(0\leq i+j\leq 4) $. Let $ \mathcal {M}(2)$ be the $ 6 \times 6$ moment matrix for $ \beta ^{(4)}$, given by $ \mathcal {M}(2)_{\mathbf {i},\,\mathbf {j}}:=\beta _{\mathbf {i}+\mathbf {j}}$, where $ \mathbf {i},\mathbf {j} \in \mathbb{Z}^2_+$ and $ \left \vert\mathbf {i}\right \vert,\left \vert\mathbf {j}\right \vert\le 2$. In this note we find concrete representing measures for $ \beta ^{(4)}$ when $ \mathcal {M}(2)$ is nonsingular; moreover, we prove that it is possible to ensure that one such representing measure is $ 6$-atomic.


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Additional Information

Raúl E. Curto
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242
Email: raul-curto@uiowa.edu

Seonguk Yoo
Affiliation: Department of Mathematics, Seoul National University, Seoul 151-742, Korea
Address at time of publication: Department of Mathematics, Inha University, Incheon 402-751, Korea
Email: seyoo73@gmail.com

DOI: https://doi.org/10.1090/proc/12698
Keywords: Nonsingular quartic binary moment problem, moment matrix extension, flat extensions, rank-one perturbations, invariance under degree-one transformations
Received by editor(s): May 12, 2014
Received by editor(s) in revised form: November 13, 2014, and December 13, 2014
Published electronically: June 30, 2015
Additional Notes: The first named author was supported by NSF Grants DMS-0801168 and DMS-1302666. The second named author was supported by the PARC postdoctoral program at Seoul National University and by the Brain Korea 21 Program of National Research Foundation of Korea (Grant number: 22A20130012598).
Communicated by: Pamela Gorkin
Article copyright: © Copyright 2015 American Mathematical Society