Solving moment problems by dimensional extension

We prove a solvability theorem for the Stieltjes moment problem on $R^d$ which is based on the multivariate Stieltjes condition $\sum_{n=1}^\infty L(x_j^n)^{-1/(2n)}=+\infty$, $j=1,\dots,d.$ This result is applied to derive a new solvability theorem for the moment problem on unbounded semi-algebraic subsets of $R^d$.

Let A be a finite subset of N^n and R[x]_A be the space of real polynomials whose monomial powers are from A. Let K be a compact basic semialgebraic set of R^n such that R[x]_A contains a polynomial that is positive on K. Denote by P_A(K) the cone of polynomials in R[x]_A that are nonnegative on K. The dual cone of P_A(K) is R_A(K), the set of all A-truncated moment sequences in R^A that admit representing measures supported in K. Our main results are: i) We study the propert...

The Moment-SOS hierarchy initially introduced in optimization in 2000, is based on the theory of the K-moment problem and its dual counterpart, polynomials that are positive on K. It turns out that this methodology can be also applied to solve problems with positivity constraints " f (x) $\ge$ 0 for all x $\in$ K " and/or linear constraints on Borel measures. Such problems can be viewed as specific instances of the " Generalized Problem of Moments " (GPM) whose list of import...

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar's Positivstellensatz over a compact basic semialgebraic set S, with a new polynomial bound on the degree of the positivity certificates. This bound involves a Lojasiewicz exponent associated to the description of S. We show that if the gradients of the active constraints are linearly independent on S (Constraint Qualification condition),this Lojasiew...

Let $\beta\equiv\beta^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(\beta)$, 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 \mu, which is r-atomic, with supp \mu$ equal to $\mathcal{V}(\mathcal{M}(n+1))$, the algebraic v...

We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group $G$ over $\R$ acting on an affine $\R$-variety $V$, we consider the induced dual action on the coordinate ring $\R[V]$ and on the linear dual space of $\R[V]$. In this setting, given an invariant closed semialgebraic subset $K$ of $V(\R)$, we study the problem of representation of invariant nonnegative polynomials on $K$ by invariant sums of squares, and the closel...

Let $\beta \equiv \{ \beta_\mathbf{i} \}_{\mathbf{i} \in \mathbb{Z}_+^d}$ be a $d$-dimensional multisequence. Curto and Fialkow, have shown that if the infinite moment matrix $M(\beta)$ is finite-rank positive semidefinite, then $\beta$ has a unique representing measure, which is $rank M(\beta)$-atomic. Further, let $\beta^{(2n)} \equiv \{ \beta_\mathbf{i} \}_{\mathbf{i} \in \mathbb{Z}_+^d, \mid \mathbf{i} \mid \leq 2n}$ be a given truncated multisequence, with associated mom...

We show that every real polynomial $f$ nonnegative on $[-1,1]^{n}$ can be approximated in the $l_{1}$-norm of coefficients, by a sequence of polynomials $\{f_{\ep r}\}$ that are sums of squares. This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as we provide a very simple and \textit{explicit} approximation sequence. Then we show that if the Moment Problem holds for a basic closed semi-algebraic set $K_S\subset\R^...

Infinite dimensional moment problems have a long history in diverse applied areas dealing with the analysis of complex systems but progress is hindered by the lack of a general understanding of the mathematical structure behind them. Therefore, such problems have recently got great attention in real algebraic geometry also because of their deep connection to the finite dimensional case. In particular, our most recent collaboration with Murray Marshall and Mehdi Ghasemi about ...

The cone of nonnegative polynomials is of fundamental importance in real algebraic geometry, but its facial structure is understood in very few cases. We initiate a systematic study of the facial structure of the cone of nonnegative polynomials $\pos$ on a smooth real projective curve $X$. We show that there is a duality between its faces and totally real effective divisors on $X$. This allows us to fully describe the face lattice in case $X$ has genus $1$. The dual cone $\po...