arXiv subjects

Lawrence Fialkow

Publications and source records attributed to Lawrence Fialkow.

3 recordsLinked to original sources

arXiv2025

The pure $Y=X^{d}$ truncated moment problem

Let $\beta \equiv\beta^{(2n)}$ be a real bivariate sequence of degree $2n$. We study the existence of representing measures for $\beta$ supported in the curve $y=x^{d}$ ($d\ge 1$) in the case when all column dependence relations in the moment matrix $M_n(\beta)$ are generated by the relation $Y=X^{d}$. We prove that the core variety of $\beta$, $\mathcal{CV}(L_{\beta})$, is nonempty (equivalently, representing measures exist) if and only if $C$, the partially defined core matrix of $\beta$, admits a positive, recursively generated completion $C[A]$. Moreover, $\mathcal{CV}(L_{\beta})$ is the entire curve $y=x^{d}$ if and only if there is a positive definite completion $C[A]$. In the remaining case, if there is a measure, it is unique and finitely atomic. For $d = 3$, we use these results to compute the core variety of $\beta$ and give new characterizations of the existence of representing measures, which complement a result of the first-named author.

math.FA↗

arXiv2018

The core variety and representing measures in the truncated moment problem

The classical Truncated Moment problem asks for necessary and sufficient conditions so that a linear functional $L$ on $\mathcal{P}_{d}$, the vector space of real $n$-variable polynomials of degree at most $d$, can be written as integration with respect to a positive Borel measure $μ$ on $\mathbb{R}^n$. We work in a more general setting, where $L$ is a linear functional acting on a finite dimensional vector space $V$ of Borel-measurable functions defined on a $T_{1}$ topological space $S$. Using an iterative geometric construction, we associate to $L$ a subset of $S$ called the \textit{core variety}, $\mathcal{CV}(L)$. Our main result is that $L$ has a representing measure $μ$ if and only if $\mathcal{CV}(L)$ is nonempty. In this case, $L$ has a finitely atomic representing measure, and the union of the supports of such measures is precisely $\mathcal{CV}(L)$. We also use the core variety to describe the facial decomposition of the cone of functionals in the dual space $V^{*}$ having representing measures. We prove a generalization of the Truncated Riesz-Haviland Theorem of Curto-Fialkow, which permits us to solve a generalized Truncated Moment Problem in terms of positive extensions of $L$. These results are adapted to derive a Riesz-Haviland Theorem for a generalized Full Moment Problem and to obtain a core variety theorem for the latter problem.

math.FA↗

arXiv2009

Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems

We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence $y$, we show that $y$ lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set $K \subseteq \re^n$ if and only if the associated Riesz functional $L_y$ is $K$-positive. For a determining set $K$, we prove that if $L_y$ is strictly $K$-positive, then $y$ admits a representing measure supported in $K$. As a consequence, we are able to solve the truncated $K$-moment problem of degree $k$ in the cases: (i) $(n,k)=(2,4)$ and $K=\re^2$; (ii) $n\geq 1$, $k=2$, and $K$ is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert's theorem on sums of squares.

math.FA↗