arXiv subjects

Lawrence A. Fialkow

Publications and source records attributed to Lawrence A. Fialkow.

5 recordsLinked to original sources

arXiv2012

Recursively determined representing measures for bivariate truncated moment sequences

A theorem of Bayer and Teichmann implies that if a finite real multisequence β= β^(2d) has a representing measure, then the associated moment matrix M_d admits positive, recursively generated moment matrix extensions M_(d+1), M_(d+2),... For a bivariate recursively determinate M_d, we show that the existence of positive, recursively generated extensions M_(d+1),...,M_(2d-1) is sufficient for a measure. Examples illustrate that all of these extensions may be required to show that βhas a measure. We describe in detail a constructive procedure for determining whether such extensions exist. Under mild additional hypotheses, we show that M_d admits an extension M_(d+1) which has many of the properties of a positive, recursively generated extension.

math.FA↗

arXiv2006

The extremal truncated moment problem

For a degree 2n real d-dimensional multisequence β^(2n) to have a representing measure, it is necessary for the associated moment matrix M(n) to be positive semidefinite and for the algebraic variety V = V(β) associated to βto satisfy rank M(n) <= card V as well as the following consistency condition: if a polynomial p vanishes on V, then p(β) = 0. We prove that for the extremal case (rank M(n) = card V), positivity of M(n) and consistency are sufficient for the existence of a (unique, rank M(n)-atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of M(n).

math.FA↗

arXiv2005

Truncated K-moment problems in several variables

Let $β\equivβ^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(β)$, and let $r:=rank \mathcal{M}(n)$. We prove that if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving moment matrix extension $\mathcal{M}(n+1)$, then $\mathcal{M}(n+1)$ has a unique representing measure μ, which is r-atomic, with supp μ$ equal to $\mathcal{V}(\mathcal{M}(n+1))$, the algebraic variety of $\mathcal{M}(n+1)$. Further, βhas an r-atomic (minimal) representing measure supported in a semi-algebraic set $K_{\mathcal{Q}}$ subordinate to a family $\mathcal{Q}% \equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]$ if and only if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving extension $\mathcal{M}(n+1)$ for which the associated localizing matrices $\mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$ are positive semidefinite $(1\leq i\leq m)$; in this case, μ(as above) satisfies supp μ\subseteq K_{\mathcal{Q}}$, and μhas precisely rank \mathcal{M}(n)-rank \mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$ atoms in $\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$, $1\leq i\leq m$.

math.FA↗

arXiv2005

Solution of the truncated hyperbolic moment problem

Let Q(x,y)=0 be an hyperbola in the plane. Given real numbers $β\equivβ^{2n)}=\{β_{ij}\}_{i,j\geq0,i+j\leq2n}$, with $β_{00}>0$, the truncated Q-hyperbolic moment problem for βentails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x,y)=0, such that $β_{ij}=\int y^{i}x^{j} dμ(0\leq i+j\leq2n)$. We prove that βadmits a Q-representing measure μ(as above) if and only if the associated moment matrix $\mathcal{M}(n)(β)$ is positive semidefinite, recursively generated, has a column relation Q(X,Y)=0, and the algebraic variety $\mathcal{V}(β)$ associated to βsatisfies $card\mathcal{V}(β)\geq\rank\mathcal{M}(n)(β)$. In this case, $rank\mathcal{M}(n)\leq2n+1$; if $rank\mathcal{M}(n)\leq2n$, then βadmits a $rank\mathcal{M}(n)$-atomic (minimal) Q-representing measure; if $rank\mathcal{M}(n)=2n+1$, then βadmits a Q-representing measure μsatisfying $2n+1\leqcard suppμ\leq2n+2$.

math.FA↗

arXiv2002

A duality proof of Tchakaloff's theorem

Tchakaloff's Theorem establishes the existence of a quadrature rule of prescribed degree relative to a positive, compactly supported measure that is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}^{d}$. Subsequent extensions were obtained by Mysovskikh and by Putinar. We provide new proofs and partial extensions of these results, based on duality techniques utilized by Stochel. We also obtain new uniqueness criteria in the Truncated Complex Moment Problem.

math.FA↗