Truncated Moment Problem Research Articles

Inverse problem for dynamical system associated with Jacobi matrices and classical moment problems

We consider Hamburger, Stieltjes and Hausdorff moment problems, that are problems of the determination of a Borel measure supported on the real axis, on the semi-axis or on the interval (0,1), from a prescribed set of moments. We propose a unified approach to these three problems based on the use of the auxiliary dynamical system with the discrete time associated with a semi-infinite Jacobi matrix. It is shown that the set of moments determines the inverse dynamic data for such a system. Using the ideas of the Boundary Control method, for every N∈N we can recover the spectral measure of N×N block of Jacobi matrix, which is a solution to a truncated moment problem. This problem is reduced to the finite-dimensional generalized spectral problem, whose matrices are constructed from moments and are connected with the well-known Hankel matrices by simple formulas. Thus the results on existence of solutions to Hamburger, Stieltjes and Hausdorff moment problems are naturally provided in terms of these matrices. We also obtain results on uniqueness of the solution of the moment problems, where as a main tool we use Krein-type equations of inverse problem.

The multidimensional truncated moment problem: Gaussian and log-normal mixtures, their Carathéodory numbers, and set of atoms

We study truncated moment sequences of distribution mixtures, especially from Gaussian and log-normal distributions and their Carath\'eodory numbers. For $\mathsf{A} = \{a_1,\dots,a_m\}$ continuous (sufficiently differentiable) functions on $\mathbb{R}^n$ we give a general upper bound of $m-1$ and a general lower bound of $\left\lceil \frac{2m}{(n+1)(n+2)}\right\rceil$. For polynomials of degree at most $d$ in $n$ variables we find that the number of Gaussian and log-normal mixtures is bounded by the Carath\'eodory numbers in \cite{didio17Cara}. Therefore, for univariate polynomials $\{1,x,\dots,x^d\}$ at most $\left\lceil\frac{d+1}{2}\right\rceil$ distributions are needed. For bivariate polynomials of degree at most $2d-1$ we find that $\frac{3d(d-1)}{2}+1$ Gaussian distributions are sufficient. We also treat polynomial systems with gaps and find, e.g., that for $\{1,x^2,x^3,x^5,x^6\}$ 3 Gaussian distributions are enough for almost all truncated moment sequences. For log-normal distributions the number is bounded by half of the moment number. We give an example of continuous functions where more Gaussian distributions are needed than Dirac delta measures. We show that any inner truncated moment sequence has a mixture which contains any given distribution.

Moment Problems in Hereditary Function Spaces

Abstract We introduce a concept of hereditary set of multi-indices, and consider vector spaces of functions generated by families associated to such sets of multi-indices, called hereditary function spaces. Existence and uniquenes of representing measures for some abstract truncated moment problems are investigated in this framework, by adapting the concept of idempotent and that of dimensional stability, and using some techniques involving C*-algebras and commuting self-adjoint multiplication operators.

A new approach to the nonsingular cubic binary moment problem

We present an alternative solution to nonsingular cubic moment problems, using techniques that are expected to be useful for higher-degree truncated moment problems. In particular, we apply the theory of recursively determinate moment matrices to deal with a case of rank-increasing moment matrix extensions.

Complex Orthogonal Polynomials and Numerical Quadrature via Hyponormality

By interpreting a truncated moment problem in two real variables in complex terms one exploits the hyponormality of the coordinate multiplier to obtain characterizations of: the moments of a positive measure, the associated complex orthogonal polynomials and the structure of the Hessenberg matrices. All reflecting the intricate structure of multivariate positive polynomials versus sums of squares or sums of hermitian squares of polynomials. The principal step in the proposed algorithm is stated in terms of a linear matrix inequality, making our approach prone for numerical applications.

The Singular Bivariate Quartic Tracial Moment Problem

The (classical) truncated moment problem, extensively studied by Curto and Fialkow, asks to characterize when a finite sequence of real numbers indexes by words in commuting variables can be represented with moments of a positive Borel measure $\mu$ on $\mathbb R^n$. In \cite{BK12} Burgdorf and Klep introduced its tracial analog, the truncated tracial moment problem, which replaces commuting variables with non-commuting ones and moments of $\mu$ with tracial moments of matrices. In the bivariate quartic case, where indices run over words in two variables of degree at most four, every sequence with a positive definite $7\times 7$ moment matrix $\mathcal M_2$ can be represented with tracial moments \cite{BK10,BK12}. In this article the case of singular $\mathcal M_2$ is studied. For $\mathcal M_2$ of rank at most 5 the problem is solved completely; namely, concrete measures are obtained whenever they exist and the uniqueness question of the minimal measures is answered. For $\mathcal M_2$ of rank 6 the problem splits into four cases, in two of which it is equivalent to the feasibility problem of certain linear matrix inequalities. Finally, the question of a flat extension of the moment matrix $\mathcal M_2$ is addressed. While this is the most powerful tool for solving the classical case, it is shown here by examples that, while sufficient, flat extensions are mostly not a necessary condition for the existence of a measure in the tracial case.

Entanglement and the truncated moment problem

We map the quantum entanglement problem onto the mathematically well-studied truncated moment problem. This yields a necessary and sufficient condition for separability that can be checked by a hierarchy of semi-definite programs. The algorithm always gives a certificate of entanglement if the state is entangled. If the state is separable, typically a certificate of separability is obtained in a finite number of steps and an explicit decomposition into separable pure states can be extracted.

The core variety of a multisequence in the truncated moment problem

Let β≡β(m)={βi}i∈Z+n,|i|≤m, β0>0, denote a real n-dimensional multisequence of finite degree m. The Truncated Moment Problem concerns the existence of a positive Borel measure μ, supported in Rn, such that(0.1)βi=∫Rnxidμ (i∈Z+n, |i|≤m). We associate to β≡β(2d) an algebraic variety in Rn called the core variety, V≡V(β). The core variety contains the support of each representing measure μ. We show that if V is nonempty, then β(2d−1) has a representing measure. Moreover, if V is a nonempty compact or determining set, then β(2d) has a representing measure. We also use the core variety to exhibit a sequence β, with positive definite moment matrix and positive Riesz functional, which fails to have a representing measure.

The truncated moment problem on [formula omitted

We find necessary and sufficient conditions for the existence of a probability measure on N0, the nonnegative integers, whose first n moments are a given n-tuple of nonnegative real numbers. The results, based on finding an optimal polynomial of degree n which is nonnegative on N0 (and which depends on the moments), and requiring that its expectation be nonnegative, generalize previous results known for n=1, n=2 (the Percus–Yamada condition), and partially for n=3. The conditions for realizability are given explicitly for n≤5 and in a finitely computable form for n≥6. We also find, for all n, explicit bounds, in terms of the moments, whose satisfaction is enough to guarantee realizability. Analogous results are given for the truncated moment problem on an infinite discrete semi-bounded subset of R.

The Division Algorithm in Sextic Truncated Moment Problems

For a degree 2n finite sequence of real numbers \(\beta \equiv \beta ^{(2n)}= \{ \beta _{00},\beta _{10},\)\(\beta _{01},\ldots , \beta _{2n,0}, \beta _{2n-1,1},\ldots , \beta _{1,2n-1},\beta _{0,2n} \}\) to have a representing measure \(\mu \), it is necessary for the associated moment matrix \(\mathcal {M}(n)\) to be positive semidefinite, and for the algebraic variety associated to \(\beta \), \(\mathcal {V}_{\beta } \equiv \mathcal {V}(\mathcal {M}(n))\), to satisfy \({\text {rank}} \mathcal {M}(n)\le {\text {card}} \mathcal {V}_{\beta }\) as well as the following consistency condition: if a polynomial \(p(x,y)\equiv \sum _{ij}a_{ij}x^{i}y^j\) of degree at most 2n vanishes on \(\mathcal {V}_{\beta }\), then the Riesz functional\(\Lambda (p) \equiv p(\beta ):=\sum _{ij}a_{ij}\beta _{ij}=0\). Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic (\(n=1\)) and quartic (\(n=2\)) moment problems. Also, positive semidefiniteness, combined with consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r) and the cardinality of the associated algebraic variety (denoted by v) are equal. For extremal sextic moment problems, verifying consistency amounts to having good representation theorems for sextic polynomials in two variables vanishing on the algebraic variety of the moment sequence. We obtain such representation theorems using the Division Algorithm from algebraic geometry. As a consequence, we are able to complete the analysis of extremal sextic moment problems.

Truncated moment problems in ℝ^2 and recursiveness

Truncated moment problems in ℝ^2 and recursiveness

Higher order asymptotic for Burgers equation and Adhesion model

This paper is focused on the study of the large time asymptotic for solutions to the viscous Burgers equation and also to the adhesion model via heat equation. Using generalization of the truncated moment problem to a complex measure space, we construct asymptotic N-wave approximate solution to the heat equation subject to the initial data whose moments exist upto the order $2n+m$ and $i$-th order moment vanishes, for $i=0, 1, 2\dots m-1$. We provide a different proof for a theorem given by Duoandikoetxea and Zuazua [3], which plays a crucial role in error estimations. In addition to this we describe a simple way to construct an initial data in Schwartz class whose $m$ moments are equal to the $m$ moments of given initial data.

Flat extensions in ∗-algebras

The main result of the paper is a flat extension theorem for positive linear functionals on ∗ * -algebras. The theorem is applied to truncated moment problems on cylinder sets, on matrices of polynomials and on enveloping algebras of Lie algebras.

A probabilistic interpretation of set-membership filtering: Application to polynomial systems through polytopic bounding

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zeroth order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P.

Recursiveness and propagation for 2-variable weighted shifts

For 1-variable weighted shifts, it is well known that given three positive numbers α0<α1<α2, we always find Stampfli's subnormal weighted shift whose first three weights are α0, α1 and α2 (we call it a type-abc completion, or shift.) On the other hand, given four positive numbers α0<α1<α2<α3, it is not always possible to find a subnormal weighted shift whose first four weights are α0, α1, α2 and α3. In this paper, we extend the notion of a type-abc shift to the 2-variable weighted shifts using a localization technique from the theory of truncated moment problems. We next study an inner propagation phenomena of a 2-variable weighted shift whose core is of type-abc. We further establish the necessary and sufficient conditions for the subnormality of one-step extensions of type-abc 2-variable weighted shifts. Finally, we study a certain invariant property under powers for type-abc 2-variable weighted shifts.

Non-extremal sextic moment problems

Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems. Also, positive semidefiniteness, combined with another necessary condition, consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r ) and the cardinality of the associated algebraic variety (denoted by v ) are equal. However, these conditions are not sufficient for non-extremal sextic or higher-order truncated moment problems. In this paper we settle three key instances of the non -extremal (i.e., r < v ) sextic moment problem, as follows: when r = 7 , positive semidefiniteness, consistency and the variety condition guarantee the existence of a 7-atomic representing measure; when r = 8 we construct two determining algorithms, corresponding to the cases v = 9 and v = + ∞ . To accomplish this, we generalize the rank-reduction technique developed in previous work, where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure.

Truncated moment problems for $J$-self-adjoint, $J$-skew-self-adjoint and $J$-unitary operators

In this paper we study truncated moment problems for $J$-self-adjoint, $J$-skew-self-adjoint and $J$-unitary operators. Conditions of the solvability are given. Some canonical solutions of the moment problems are constructed. As a by-product, some extension results for $J$-skew-symmetric and $J$-isometric operators are obtained.

Local moment problem

AbstractThe work is devoted to the local moment problem, which consists in finding of non‐decreasing functions on the real axis having given first 2n + 1, n ≥ 0, power moments on the whole axis and also 2m + 1 first power moments on a certain finite axis interval. Considering the local moment problem as a combination of the Hausdorff and Hamburger truncated moment problems we obtain the conditions of its solvability and describe the class of its solutions with minimal number of growth points if the problem is solvable. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Some Propositions on Generalized Nevanlinna Functions of the ClassNk

Some propositions on the generalized Nevanlinna functions are derived. We indicate mainly that (1) the negative inertia index of a Hermitian generalized Loewner matrix generated by a generalized Nevanlinna function in the classNκdoes not exceedκ. This leads to an equivalent definition of a generalized Nevanlinna function; (2) if a generalized Nevanlinna function in the classNκhas a uniform asymptotic expansion at a real pointαor at infinity, then the negative inertia index of the Hankel matrix constructed with the partial coefficients of that asymptotic expansion does not exceedκ. Also, an explicit formula for the negative index of a real rational function is given by using relations among Loewner, Bézout, and Hankel matrices. These results will provide first tools for the solution of the indefinite truncated moment problems together with the multiple Nevanlinna-Pick interpolation problems in the classNκbased on the so-called Hankel vector approach.

Cubic column relations in truncated moment problems

For the truncated moment problem associated to a complex sequence γ(2n)={γij}i,j∈Z+,i+j⩽2n to have a representing measure μ, it is necessary for the moment matrix M(n) to be positive semidefinite, and for the algebraic variety Vγ to satisfy rankM(n)⩽cardVγ as well as a consistency condition: the Riesz functional vanishes on every polynomial of degree at most 2n that vanishes on Vγ. In previous work with L. Fialkow and H.M. Möller, the first named author proved that for the extremal case (rankM(n)=cardVγ), positivity and consistency are sufficient for the existence of a representing measure. In this paper we solve the truncated moment problem for cubic column relations in M(3) of the form Z3=itZ+uZ¯ (u,t∈R); we do this by checking consistency. For (u,t) in the open cone determined by 0<|u|<t<2|u|, we first prove that the algebraic variety has exactly 7 points and rankM(3)=7; we then apply the above mentioned result to obtain a concrete, computable, necessary and sufficient condition for the existence of a representing measure.