On Truncated Matricial Stieltjes Type Moment Problems

We study two slightly different versions of truncated generalized matricial moment problems of Stieltjes type. A central topic is the construction and investigation of distinguished solutions of both moment problems under consideration. These solutions turn out to be nonnegative Hermitian Borel measures on the real axis which are concentrated on a finite number of points. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials constructed from a nonnegative Hermitian block Hankel matrix.     

An Idempotent Approach to Truncated Moment Problems

We present a new approach to truncated and full moment problems, via idempotent elements with respect to associated square positive Riesz functionals. The existence of representing measures for such functionals is characterized via some intrinsic conditions.     

Hamburger Moment Sequences in Combinatorics

We show that many well-known counting coefficients in combinatorics are Hamburger moment sequences in certain unified approaches and that Hamburger moment sequences are infinitely convex. We introduce the concept of the q-Hamburger moment sequence of polynomials and present some examples of such sequences of polynomials. We also suggest some problems and conjectures.     

完全不确定Hamburger矩阵矩量问题的有限阶解

该文描述带有矩量序列{v_m}_0~∞■C~(q×q)的完全不确定Hamburger矩阵矩量问题:v_m=integral from n=-∞to∞x~m dρ(x),m=0,1,…的有限阶解,即该问题的那些解ρ,使得C~(q×q)-值多项式的线性空间P在对应的空间L~2(R,dρ/E(x))内稠密,这里E(x)为在实轴R上取正值的某个数值多项式.作为预备知识,作者考虑所谓广义Akhiezer插值的矩阵变种与它的相关矩阵矩量问题之间的一种关系.     

Truncated VSV Solutions to Symmetric Rank-Deficient Problems

Symmetric VSV decompositions are new rank-revealing decompositions that exploit and preserve symmetry. Truncated VSV solutions are stabilized solutions computed by neglecting blocks in the VSV decomposition with small norm. We compare the truncated VSV solutions with truncated SVD solutions and give perturbation bounds for the VSV solutions. Numerical examples illustrate our results.     

The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

On V-Extremal Solutions of the Moment Problem

We study the density problem for a set of polynomials in the space , where is a measure with finite moments. The approach to this problem is based on methods of the theory of moments, which allows one to formulate sufficient conditions in terms of Nevanlinna functions.     

Solution of the Truncated Parabolic Moment Problem

Given real numbers with ₀₀ >0 , the truncated parabolic moment problem for entails finding necessary and sufficient conditions for the existence of a positive Borel measure , supported in the parabola p(x, y) = 0, such that We prove that admits a representing measure (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated and has a column relation p(X, Y) = 0, and the algebraic variety () associated to satisfies card In this case, admits a rank -atomic (minimal) representing measure.Submitted: August 25, 2003     

Solution of the Truncated Hyperbolic Moment Problem

Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β ⁽²ⁿ⁾={ β _ij } _{i,j ≥ 0,i+j ≤ 2n} , with β₀₀ > 0, the truncated Q-hyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x, y) = 0, such that We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety associated to β satisfies card In this case, if then β admits a rank -atomic (minimal) Q-representing measure; if then β admits a Q-representing measure μ satisfying      

Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation

We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz ² ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g⁽²ⁿ⁾:g₀₀,g₀₁,g₁₀,?,g_0,2n,g_1,2n-1,?,g_2n-1,1,g_2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that g_ij=ò[`(z)]<sup>i</sup>z<sup>j</sup> dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ <sup>2</sup>, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number g<sub>n,n+1</sub> \gamma_{n,n+1} satisfying g<sub>n+1,n</sub> o [`(g)]_n,n+1=Ag_n,n-1+Bg_n,n+Cg_n+1,n-1+Dg_n,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z ^k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

Tight Frames of Polynomials and the Truncated Trigonometric Moment Problem

A simple parametrization is given for the set of positive measures with finite support on the circle group T that are solutions of the truncated trigonometric moment problem: where the parameters are, up to nonzero multiplicative constants, the polynomials whose roots all have a modulus less than one. This result is then used to characterize, on a certain natural Hilbert space of polynomials associated with the problem, all finite "weighted" tight frames of evaluation polynomials. Finally, a new and simple way of parametrizing the whole set of positive Borel measures on T, solutions of the given moment problem is deduced, via a limiting argument.     

On the Stability of Solutions to the Moment System of Nonequilibrium Thermodynamics

The paper is devoted to the study of the asymptotic behavior of solutions to linearized systems of equations of the so-called nonequilibrium thermodynamic moments.     

On the Global Stability of Solutions of Moment Systems in Nonequilibrium Thermodynamics

In this paper, we study the linearization of the Cauchy problem and the mixed problem for the system of Grad--Hermite moments in nonequilibrium thermodynamics in the neighborhood of the equilibrium state. Stability conditions for solutions of the Cauchy problem are proved as a generalization of the classical Hermite--Biller theorem on stable polynomials. For the mixed problem, we prove an analog of the Vishik--Lyusternik theorem on small singular perturbations of general elliptic problems. The last observation allows us to introduce the Shapiro--Lopatinskii condition, which implies the well-posedness of the mixed problem.     

On Spurious Solutions of Singular Perturbation Problems

A study is made of several nonlinear boundary-value problems of singular perturbation type for which a straightforward application of boundary-layer theory leads to spurious solutions. It is shown that these problems can be treated successfully by a slight modification of the method of matched asymptotic expansions. The analysis leads to several novel features which are not present in routine singular perturbation problems.     

Moment problem of Hamburger,hierarchies of integrable systems,and the positivity of tau-functions

A moment problem of Hamburger is studied to find a parametric Stieltjes measure from given moments. It is shown that if a deformation, or a dynamics, of moments is governed by a hierarchy of a Kac-van Moerbeke system, then the Stieltjes measure can be constructed explicitly by integrating a hierarchy of Moser's nonlinear dynamical system. The positivity of tau-functions is related to the existence of the Stieltjes measure at a deep level.     

Partially Commutative Moment Problems

Partially commutative tensor algebras occur naturally in the algebraic formulation of Wightman field theory. A state on an algebra of this type leads via GNS-construction to a partially commutative family of hermitean operators on Hilbert space. We discuss the question when these operators can be extended to self adjoint operators preserving the commutation properties and state a necessary and sufficient condition for the existence of such an extension in terms of a positivity property of the state.     

The Favoured Solutions of the Moment Problem

The sequence of Catalan numbers is known to be the moments of the Wigner law. A characterization of this sequence is given in terms of some determinants and the moment sequence of the q-Gaussian law is characterized in a similar fashion.