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The Core Variety of a Multisequence in the Truncated Moment Problem

Lawrence Fialko
SUNY, New Paltz


Let $\beta \equiv \beta(m) = {\beta_i }_{i\in Z_+^n,|i| \le m}, \beta_0 > 0$ denote a real n-dimensional multisequence of finite degree $m$. The Truncated Moment Problem concerns the existence of a positive Borel measure $\mu$, supported in $R^n$ , such that $$ \beta_i =\int_{R^n} x^i d\mu (i\in Z_+^n, |i| \le m). $$ We associate to $\beta \equiv \beta_^{(2d)}$ an algebraic variety in $R^n$ called the core variety, $V \equiv V(\beta)$. The core variety contains the support of each representing measure $\mu$ . We show that if $V$ is nonempty, then $\beta^{(2d-1)}$ has a representing measure. Moreover, if $V$ is a nonempty compact or determining set, then $\beta^{(2d)}$ has a representing measure. We also use the core variety to exhibit a sequence $\beta$, with positive definite moment matrix and positive Riesz functional, which fails to have a representing measure.

Tuesday, January 13, 2015
11:00AM AP&M 2402