Complex Moment Problem Research Articles

Finite-Dimensional Completion for the Matrix-Valued Truncated Complex Moment Problem

Finite-Dimensional Completion for the Matrix-Valued Truncated Complex Moment Problem

The inner structure of the block Jacobi type matrix related to the complex moment problem with the measure supported on the second order curve

The inner structure of the block Jacobi type matrix related to the complex moment problem with the measure supported on the second order curve

The complex moment problem of Dirichlet type

Solving a moment problem means, roughly speaking, to find a spectral representation of a given data. This is usually understood as a typical inverse spectral problem, also because of the traditional connection between moments and operators; the latter is the object of special concern in the present paper.

Dirichlet-type spaces on the unit ball and joint 2-isometries

We obtain a formula that relates the spherical moments of the multiplication tuple on a Dirichlet-type space to a complex moment problem in several variables. This can be seen as the ball-analogue of a formula originally invented by Richter in [23]. We capitalize on this formula to study Dirichlet-type spaces on the unit ball and joint 2-isometries.

The quintic complex moment problem

Let $\gamma^{(m)} \equiv \{ \gamma_{ij} \}_{0 \leq i +j \leq m}$ be a given complex-valued sequence. The truncated complex moment problem (TCMP in short) involves determining necessary and sufficient conditions for the existence of a positive Borel measure $\mu$ on $\mathbb{C}$ (called a representing measure for $\gamma^{(m)}$) such that $\gamma_{ij} = \int \overline{z}^i z^j d\mu$ for $0 \leq i +j \leq m$. The TCMP has been completely solved only when $m= 1, 2, 3, 4$. We provide in this paper a concrete solution to the quintic TCMP (that is, when $m = 5$). We also study the cardinality of the minimal representing measure. Based on the bivariate recurrences sequences's properties with some Curto-Fialkow's results, our method intended to be useful for all odd-degree moment problems.

Charges solve the truncated complex moment problem

Let [Formula: see text], with [Formula: see text] and [Formula: see text], be a given complex-valued sequence. The complex moment problem (respectively, the general complex moment problem) associated with [Formula: see text] consists in determining necessary and sufficient conditions for the existence of a positive Borel measure (respectively, a charge) [Formula: see text] on [Formula: see text] such that [Formula: see text] In this paper, we investigate the notion of recursiveness in the two variable case. We obtain several useful results that we use to deduce new necessary and sufficient conditions for the truncated complex moment problem to admit a solution. In particular, we show that the general complex moment problem always has a solution. A concrete construction of the solution and an illustrating example are also given.

A general fibre theorem for moment problems and some applications

The fibre theorem [12] for the moment problem on closed semi-algebraic subsets of Rd is generalized to finitely generated real unital algebras. As an application two new theorems on the rational multidimensional moment problem are proved. Another application is a characterization of moment functionals on the polynomial algebra R[x1,..., xd] in terms of extensions. Finally, the fibre theorem and the extension theorem are used to reprove basic results on the complex moment problem due to Stochel and Szafraniec [13] and Bisgaard [2].

The Cubic Complex Moment Problem

Let \({s = \{s_{jk}\}_{0 \leq j+k \leq 3}}\) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure \({\sigma}\) on \({\mathbb{C}}\) (called a representing measure for s) such that \({s_{jk} = \int_{\mathbb{C}}\bar{z}^j z^k d\sigma(z)}\) for \({0 \leq j + k \leq 3}\). Put $$\Phi = \left(\begin{array}{lll} s_{00} & s_{01} & s_{10} \\s_{10} & s_{11} & s_{20} \\s_{01} & s_{02} & s_{11}\end{array}\right), \quad \Phi_z = \left(\begin{array}{lll}s_{01} & s_{02} & s_{11} \\s_{10} & s_{12} & s_{21} \\s_{02} & s_{03} & s_{12}\end{array} \right)\quad {\rm and}\quad\Phi_{\bar{z}} = (\Phi_z)^*.$$ If \({\Phi \succ 0}\), then the commutativity of \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) is necessary and sufficient for the existence a 3-atomic representing measure for s. If \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set \({K \subseteq \mathbb{C}}\) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure \({\sigma}\) which satisfies \({{\rm supp} \sigma \cap K \neq \emptyset}\) or \({{\rm supp} \sigma \subseteq K}\). The cases when \({K = \overline{\mathbb{D}}}\) and \({K = \mathbb{T}}\) are considered in detail.

Riesz–Haviland criterion for incomplete data

The aim of this paper is to analyze a “support-free” version of the Riesz–Haviland theorem proved recently by the present authors, which characterizes truncations of the complex moment problem via positivity condition on appropriate families of polynomials in z and z ¯ . The attention is focused on modifications of the positivity condition as well as the assumption on admissible truncations. The former results in truncations for which the corresponding “support-free” Riesz–Haviland condition locates a representing measure on the distinguished subset of the complex plane, while the latter effects a non-integral variant of the Riesz–Haviland theorem.

On the two-dimensional moment problem

In this paper we obtain an algorithm towards solving the two-dimensional moment problem. This algorithm gives the necessary and sufficient conditions for the solvability of the moment problem. It is shown that all solutions of the moment problem can be constructed using this algorithm. In a consequence, analogous results are obtained for the complex moment problem.

Complex Moment Problems and Recursive Relations of Fibonacci Type

The complex moment problem for a sequence γ(2n) = {γ ij }0≤i,j≤n has been studied by Curto-Fialkow, where positivity and extension properties of the moment matrix M(n) = M(n)(γ) (γ ≡ γ(2n)) are involved, for guaranteeing the existence of representing measure. But it was showed that positivity and recursiveness are not sufficient in order to have a representing measure for γ. Here we combine our techniques based on the Fibonacci sequences’s properties with some Curto-Fialkow’s results to obtain sufficient conditions for insuring that γ is a truncated moment sequence. We focus ourself on the case when rank M(n) ≤ n + 1, and finally we stretch our exploration to the finite-rank infinite positive moment matrix.

Flat Extensions of Nonsingular Moment Matrices

In this paper we consider the truncated complex moment problem suggested by Curto and Fialkow. First, we give another computing proof for a solution of the nonsingular quadratic moment problem. Then we consider the nonsingular quartic moment problem and give some partial solutions.

COMPLEX MOMENT MATRICES VIA HALMOS-BRAM AND EMBRY CONDITIONS

By considering a bridge between Bram-Halmos and Embry characterizations for the subnormality of cyclic operators, we extend the Curto-Fialkow and Embry truncated complex moment problem, and solve the problem finding the finitely atomic representing measure <TEX>${\mu}$</TEX> such that <TEX>${\gamma}_{ij}={\int}\bar{z}^iz^jd{\mu},\;(0{\le}i+j{\le}2n,\;|i-j|{\le}n+s,\;0{\le}s{\le}n);$</TEX> the cases of s = n and s = 0 are induced by Bram-Halmos and Embry characterizations, respectively. The former is the Curto-Fialkow truncated complex moment problem and the latter is the Embry truncated complex moment problem.

On the complex moment problem

AbstractThe article is devoted to the solution of the infinite‐dimensional variant of the complex moment problem, and to the uniqueness of the solution. The main approach is illustrated for the best explanation on the one‐dimensional case. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A connection between the moment problem and the subnormal completion problem

In this note we give a connection between the truncated complex moment problem and the subnormal completion problem for the unilateral weighted shifts.

Reconstruction of orientations of a moving protein domain from paramagnetic data

We study the inverse problem of determining the position of the moving C-terminal domain in a metalloprotein from measurements of its mean paramagnetic tensor . The latter can be represented as a finite sum involving the corresponding magnetic susceptibility tensor χ and a finite number of rotations. We obtain an optimal estimate for the maximum probability that the C-terminal domain can assume a given orientation, and we show that only three rotations are required in the representation of , and that in general two are not enough. We also investigate the situation in which a compatible pair of mean paramagnetic tensors is obtained. Under a mild assumption on the corresponding magnetic susceptibility tensors, justified on physical grounds, we again obtain an optimal estimate for the maximum probability that the C-terminal domain can assume a given orientation. Moreover, we prove that only ten rotations are required in the representation of the compatible pair of mean paramagnetic tensors, and that in general three are not enough. The theoretical investigation is concluded by a study of the coaxial case, when all rotations are assumed to have a common axis. Results are obtained via an interesting connection with another inverse problem, the quadratic complex moment problem. Finally, we describe an application to experimental NMR data.

Embry truncated complex moment problem

Let T be a cyclic subnormal operator on a Hilbert space H with cyclic vector x 0 and let γ ij :=( T * i T j x 0, x 0), for any i,j∈ N∪{0} . The Bram–Halmos’ characterization for subnormality of T involved a moment matrix M( n). In a parallel approach, we construct a moment matrix E( n) corresponding to Embry’s characterization for subnormality of T. We discuss the relationship between M( n) and E( n) via the full moment problem. Next, given a collection of complex numbers γ≡{ γ ij } (0⩽ i+ j⩽2 n, | i− j|⩽ n) with γ 00>0 and γ ji= γ ̄ ij , we consider the truncated complex moment problem for γ; this entails finding a positive Borel measure μ supported in the complex plane C such that γ ij=∫ z ̄ iz j dμ(z) . We show that this moment problem can be solved when E( n)⩾0 and E( n) admits a flat extension E( n+ k), where k=1 when n is odd and k=2 when n is even.

Two variable subnormal completion problem

Given m > 0 and a finite collection of pairs of positive numbers C = {(α 1 (k),α 2 (k))}‖k‖≤m (‖k‖:= k 1 + k 2 ). The two variable subnormal completion problem is to find necessary and sufficient conditions to guarantee the existence of a two variable subnormal weighted shift whose initial weighted are given by C. Curto and Fialkow solved this problem in the case m = 1, using the solution of truncated complex moment problem. This paper obtained the representing measure in detail.

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 Rd. 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.

The quadratic moment problem for the unit circle and unit disk

For the quadratic complex moment problem $$\gamma _{ij} = \int {\bar z^i } z^j d\mu \left( {0 \leqslant i + j \leqslant 2} \right)$$ , we obtain necessary and sufficient conditions for the existence of representing measures μ supported in the unit circleT or in the closed unit disk $$\bar D$$ . We explicitly construct all finitely atomic representing measures supported inT or $$\bar D$$ which have the fewest atoms possible. For the quadratic $$\bar D$$ -moment problem in which the moment matrixM(1) is positive and invertible, there exists an ellipse eɛD such that the minimal (3-atomic) representing measures are supported in the complement of the interior region of e. Finally, we apply these results to obtain information on the location of the zeros of certain cubic polynomials.