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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Concrete solution to the nonsingular quartic binary moment problem
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by Raúl E. Curto and Seonguk Yoo PDF
Proc. Amer. Math. Soc. 144 (2016), 249-258 Request permission


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 $\left |\mathbf {i}\right |,\left |\mathbf {j}\right |\le 2$. 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
  • MR Author ID: 53500
  • Email:
  • 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
  • MR Author ID: 1048067
  • Email:
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 249-258
  • MSC (2010): Primary 47A57, 44A60, 42A70, 30A05; Secondary 15A15, 15-04, 47N40, 47A20
  • DOI:
  • MathSciNet review: 3415593