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 | References | Similar Articles | Additional Information

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 $\left |\mathbf {i}\right |,\left |\mathbf {j}\right |\le 2$. 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:
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

MR Author ID:
1048067

Email:
seyoo73@gmail.com

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