• Kyungpook Mathematical Journal
• /
• v.62 no.1
• /
• pp.107-118
• /
• 2022

In this note we observe that any truncated multi-index sequence has an interpolating measure supported in Euclidean space. It is well known that the consistency of a truncated moment sequence is equivalent to the existence of an interpolating measure for the sequence. When the moment matrix of a moment sequence is nonsingular, the sequence is naturally consistent; a proper perturbation to a given moment matrix enables us to confirm the existence of an interpolating measure for the moment sequence. We also illustrate how to find an explicit form of an interpolating measure for some cases.

• Korean Journal of Mathematics
• /
• v.29 no.4
• /
• pp.741-747
• /
• 2021

The Cayley-Bacharach theorem says that every cubic curve on an algebraically closed field that passes through a given 8 points must contain a fixed ninth point, counting multiplicities. Ren et al. introduced a concrete formula for the ninth point in terms of the 8 points [4]. We would like to consider a different approach to find the ninth point via the theory of truncated moment problems. Various connections between algebraic geometry and truncated moment problems have been discussed recently; thus, the main result of this note aims to observe an interplay between linear algebra, operator theory, and real algebraic geometry.

• Kyungpook Mathematical Journal
• /
• v.45 no.2
• /
• pp.241-247
• /
• 2005

Let ${\gamma}{\equiv}\left{{\gamma}_{ij}\right}(0{\leq}i+j{\leq}2n)$ be a collection of complex numbers with ${\gamma}_{00}>0$ and ${\gamma}_{ji}={\bar{\gamma}}_{ij}$. The truncated complex moment problem for ${\gamma}$ entails finding a positive Borel measure ${\mu}$ supported in the complex plane ${\mathbb{C}}$ such that ${\gamma}_{ij}={\int}{\bar{z}}^{i}z^jd{\mu}(z)(0{\leq}i+j{\leq}2n)$. We solve this truncated moment problem with data in a circle and discuss the behavior of data in an extended moment matrix.

• East Asian mathematical journal
• /
• v.40 no.1
• /
• pp.25-36
• /
• 2024

Full univariate moment problems have been studied using continued fractions, orthogonal polynomials, spectral measures, and so on. On the other hand, the truncated moment problem has been mainly studied through confirming the existence of the extension of the moment matrix. A few articles on the multivariate moment problem implicitly presented about some results of this note, but we would like to rearrange the important results for the existence of a representing measure of a moment sequence. In addition, new techniques with orthogonal polynomials will be introduced to expand the means of studying truncated moment problems.

• East Asian mathematical journal
• /
• v.36 no.1
• /
• pp.61-71
• /
• 2020

Up to the present day, the best solution we can get to the truncated moment problem (TMP) is probably the Flat Extension Theorem. It says that if the corresponding moment matrix of a moment sequence admits a rank-preserving positive extension, then the sequence has a representing measure. However, constructing a flat extension for most higher-order moment sequences cannot be executed easily because it requires to allow many parameters. Recently, the author has considered various decompositions of a moment matrix to find a solution to TMP instead of an extension. Using a new approach with the Hadamard product, the author would like to introduce more techniques related to moment matrix decompositions.

• Communications of the Korean Mathematical Society
• /
• v.35 no.1
• /
• pp.137-149
• /
• 2020

Let γ⁽ⁿ⁾ ≡ {γ_ij} (0 ≤ i+j ≤ 2n, |i-j| ≤ n) be a sequence in the complex number set ℂ and let E (n) be the Embry truncated moment matrices corresponding from γ⁽ⁿ⁾. For an odd number n, it is known that γ⁽ⁿ⁾ has a rank E (n)-atomic representing measure if and only if E(n) ≥ 0 and E(n) admits a flat extension E(n + 1). In this paper we suggest a related problem: if E(n) is positive and nonsingular, does E(n) have a flat extension E(n + 1)? and give a negative answer in the case of E(3). And we obtain some necessary conditions for positive and nonsingular matrix E (3), and also its sufficient conditions.

• Kyungpook Mathematical Journal
• /
• v.54 no.2
• /
• pp.157-163
• /
• 2014

For a finite subset ${\Lambda}$ of $\mathbb{N}_0{\times}\mathbb{N}_0$, where $\mathbb{N}_0$ is the set of nonnegative integers, we say that a complex data ${\gamma}_{\Lambda}:=\{{\gamma}_{ij}\}_{(ij){\in}{\Lambda}}$ in the unit disc $\mathbf{D}$ of complex numbers has a cyclic subnormal completion if there exists a Hilbert space $\mathcal{H}$ and a cyclic subnormal operator S on $\mathcal{H}$ with a unit cyclic vector $x_0{\in}\mathcal{H}$ such that ${\langle}S^{*i}S^jx_0,x_0{\rangle}={\gamma}_{ij}$ for all $i,j{\in}\mathbb{N}_0$. In this note, we obtain some sufficient conditions for a cyclic subnormal completion of ${\gamma}_{\Lambda}$, where ${\Lambda}$ is a finite subset of $\mathbb{N}_0{\times}\mathbb{N}_0$.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.18 no.3
• /
• pp.291-302
• /
• 2005

Due to the importance of the parameter in structural response, the uncertain elastic modulus was located at the center of stochastic analysis, where the response variability caused by the uncertain system parameters is pursued. However when we analyze the so-called stochastic systems, as many parameters as possible must be included in the analysis if we want to obtain the response variability that can reach a true one, even in an approximate sense. In this paper, a formulation to determine the statistical behavior of in-plane structures due to multiple uncertain material parameters, i.e., elastic modulus and Poisson's ratio, is suggested. To this end, the polynomial expansion on the coefficients of constitutive matrix is employed. In constructing the modified auto-and cross-correlation functions, use is made of the general equation for n-th moment. For the computational purpose, the infinite series of stochastic sub-stiffness matrices is truncated preserving required accuracy. To demons4rate the validity of the proposed formulation, an exemplary example is analyzed and the results are compared with those obtained by means of classical Monte Carlo simulation, which is based on the local averaging scheme.