The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the S-moment problem and its dual counterpart: polynomials that are positive on S. It turns out that this methodology can also be used to solve problems with positivity constraints ‘f(x) ≥ 0 for all $\mathbf{x}\in S$’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the generalized moment problem (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.

In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers $p< q$ do the equalities $T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$, $k=p, q$, imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if $p$ is odd and $q$ is even, and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on $C^*$-algebras is also provided.

This paper considers logarithmic asymptotics of tails of randomly stopped sums. The stopping is assumed to be independent of the underlying random walk. First, finiteness of ordinary moments is revisited. Then the study is expanded to more general asymptotic analysis. Results are applicable to a large class of heavy-tailed random variables. The main result enables one to identify if the asymptotic behaviour of a stopped sum is dominated by its increments or the stopping variable. As a consequence, new sufficient conditions for the moment determinacy of compounded sums are obtained.

In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares $\sum{\mathbb{R}{{\left[ \underline{X} \right]}^{2}}}$ in the polynomial ring $\mathbb{R}\left[ \underline{X} \right]\,:=\,\mathbb{R}\left[ {{X}_{1}},\,.\,.\,.\,,\,{{X}_{n}} \right]$ in the topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$, the cone consisting of all polynomials that are non-negative on the hypercube ${{\left[ -1,\,1 \right]}^{n}}$. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $~{{\ell }_{1}}$-seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theorem from 2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of $2d$-powers, proving, in particular, that for any integer $d\,\ge \,1$, the closure of the cone of sums of $2d$-powers $\sum \mathbb{R}{{\left[ \underline{X} \right]}^{2d}}$ in the $\mathbb{R}\left[ \underline{X} \right]$ topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$.

We study the class of logarithmic skew-normal (LSN) distributions. They have heavy tails; however, all their moments of positive integer orders are finite. We are interested in the problem of moments for such distributions. We show that the LSN distributions are all nonunique (moment-indeterminate). Moreover, we explicitly describe Stieltjes classes for some LSN distributions; they are families of infinitely many distributions, which are different but have the same moment sequence as a fixed LSN distribution.

We complete the investigation of growth properties of analytic functions connected with the Nevanlinna parametrization of the solutions of an indeterminate strong Hamburger moment problem.

We prove convergence of diagonal multipoint Padé approximants of Stieltjes-type functions when a certain moment problem is determinate. This is used for the study of the convergence of Fourier–Padé and nonlinear Fourier–Padé approximants for such type of functions.

We employ techniques from the theory of disintegration of measures to study the Lifting Problem for commuting $n$-tuples of subnormal weighted shifts. We obtain a new necessary condition for the existence of a lifting, and generate new pathology associated with bringing together the Berger measures associated to each individual weighted shift. For subnormal $2$-variable weighted shifts, we then find the precise relation between the Berger measure of the pair and the Berger measures of the shifts associated to horizontal rows and vertical columns of weights.

In this paper, we describe a class of Wiener functionals that are ‘indeterminate by their moments’, that is, whose distributions are not uniquely determined by their moments. In particular, it is proved that the integral of a geometric Brownian motion is indeterminate by its moments and, moreover, shown that previous proofs of this result are incorrect. The main result of this paper is based on geometric inequalities in Gauss space and on a generalization of the Krein criterion due to H. L. Pedersen.

In this paper we investigate when negative definite functions on commutative hypergroups satisfy the Schoenberg criterion.

Let F be a probability distribution function with density f. We assume that (a) F has finite moments of any integer positive order and (b) the classical problem of moments for F has a nonunique solution (F is M-indeterminate). Our goal is to describe a , where h is a ‘small' perturbation function. Such a class S consists of different distributions Fε (fε is the density of Fε) all sharing the same moments as those of F, thus illustrating the nonuniqueness of F, and of any Fε, in terms of the moments. Power transformations of distributions such as the normal, log-normal and exponential are considered and for them Stieltjes classes written explicitly. We define a characteristic of S called an index of dissimilarity and calculate its value in some cases. A new Stieltjes class involving a power of the normal distribution is presented. An open question about the inverse Gaussian distribution is formulated. Related topics are briefly discussed.

The paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real $n$-space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways.

A recent paper by Lin and Stoyanov is devoted to the moment problem for geometrically compounded sums. The aim of this note is to provide affirmative answers to their conjectures.

We deal with compound geometric sums of independent positive random variables and study the moment problem for the distributions of such sums (the Stieltjes moment problem). We find conditions under which the distributions are uniquely determined by their moments. We also treat related topics, including the Hamburger moment problem involving random variables on the whole real line. Some conjectures are outlined.

In this note we examine the relationships between a subnormal shift, the measure its moment sequence generates, and those of a large family of weighted shifts associated with the original shift. We examine the effects on subnormality of adding a new weight or changing a weight. We also obtain formulas for evaluating point mass at the origin for the measure associated with the shift. In addition, we examine the relationship between the measure associated with a subnormal shift and those of a family of shifts substantially different from the original shift.

A generalization of Schmüdgen’s Positivstellensatz is given which holds for any basic closed semialgebraic set in ${{\mathbb{R}}^{n}}$ (compact or not). The proof is an extension of Wörmann’s proof.

A preordering $T$ is constructed in the polynomial ring $A=\mathbb{R}[{{t}_{1}},{{t}_{2}},...]$ (countablymany variables) with the following two properties: (1) For each $f\,\in \,A$ there exists an integer $N$ such that $-\,N\,\le \,f\left( p \right)\,\le \,N$ holds for all $P\in \text{Spe}{{\text{r}}_{T}}(A)$. (2) For all $f\,\in \,A$, if $N+f,N-f\in T$ for some integer $N$, then $f\,\in \,\mathbb{R}$. This is in sharp contrast with the Schmüdgen-Wörmann result that for any preordering $T$ in a finitely generated $\mathbb{R}$-algebra $A$, if property (1) holds, then for any $f\in A,f>0\,\text{on}\,\text{Spe}{{\text{r}}_{T}}(A)\Rightarrow f\in T$. Also, adjoining to $A$ the square roots of the generators of $T$ yields a larger ring $C$ with these same two properties but with $\sum{{{C}^{2}}}$ (the set of sums of squares) as the preordering.

We use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.