Moment Problem Research Articles

Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation

We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem on the unbounded closed interval [0,+∞). Necessary and sufficient conditions for the existence and uniqueness of the solution are pointed out. Operator-valued moment problems and a scalar-valued moment problem are solved.

On the Linear Recurrence of (Generalized) Hybrid Numbers Sequences and Moment Problems

On the Linear Recurrence of (Generalized) Hybrid Numbers Sequences and Moment Problems

On a difference equation with holomorphic coefficients generated by a hexagon

Let D be a hexagon with two straight and four right angles. We consider a sevenelement difference equation with holomorphic coefficients generated by this hexagon. A method for regularizing this equation is proposed. The solution is sought in the class of functions holomorphic outside the "half"of the boundary ᲛD and vanishing at infinity. It is represented as a Cauchy-type integral over the "half"of the boundary with an unknown density. Applications to the problem of moments for entire functions of exponential type (e.f.e.t.) are indicated.

On the matricial truncated moment problem. II

We continue the study of truncated matrix-valued moment problems begun in [12]. Let q∈N. Suppose that (X,X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into Hq, the Hermitian q×q matrices. A linear functional Λ on E is called a moment functional if there exists a positive Hq-valued measure μ on (X,X) such that Λ(F)=∫X〈F,dμ〉 for F∈E.In this paper a number of special topics on the truncated matricial moment problem are treated. We restate a result from [11] to obtain a matricial version of the flat extension theorem. Assuming that X is a compact space and all elements of E are continuous on X we characterize moment functionals in terms of positivity and obtain an ordered maximal mass representing measure for each moment functional. The set of masses of representing measures at a fixed point and some related sets are studied. The class of commutative matrix moment functionals is investigated. We generalize the apolar scalar product for homogeneous polynomials to the matrix case and apply this to the matricial truncated moment problem.

Construction of Solutions of Toda Lattices by the Classical Moment Problem

Construction of Solutions of Toda Lattices by the Classical Moment Problem

The general divisor problem of higher moments of coefficients attached to the Dedekind zeta function

The general divisor problem of higher moments of coefficients attached to the Dedekind zeta function

Moment Problems and Integral Equations

The first part of this work provides explicit solutions for two integral equations; both are solved by means of Fourier transform. In the second part of this paper, sufficient conditions for the existence and uniqueness of the solutions satisfying sandwich constraints for two types of full moment problems are provided. The only given data are the moments of all positive integer orders of the solution and two other linear, not necessarily positive, constraints on it. Under natural assumptions, all the linear solutions are continuous. With their value in the subspace of polynomials being given by the moment conditions, the uniqueness follows. When the involved linear solutions and constraints are positive, the sufficient conditions mentioned above are also necessary. This is achieved in the third part of the paper. All these conditions are written in terms of quadratic expressions.

The Truncated Moment Problem on Reducible Cubic Curves I: Parabolic and Circular Type Relations

In this article we study the bivariate truncated moment problem (TMP) of degree 2k on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved in Zalar (Linear Algebra Appl 649:186–239, 2022. https://doi.org/10.1016/j.laa.2022.05.008), while the degree 6 cases in Yoo (Integral Equ Oper Theory 88:45–63, 2017). Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle y(ay+x2+y2)=0,a∈R\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y(ay+x^2+y^2)=0, a\\in {\\mathbb {R}}{\\setminus } \\{0\\}$$\\end{document}, and a union of a line and a parabola y(x-y2)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y(x-y^2)=0$$\\end{document}. In both cases we also determine the number of atoms in a minimal representing measure.

On the truncated matricial moment problem. I

This paper is about the general truncated matrix-valued moment problem. Let Hq denote the complex Hermitian q×q-matrices, q∈N. Suppose that (X,X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into Hq. A linear functional Λ on E is called a moment functional if there exists a positive Hq-valued measure μ on (X,X) such that Λ(F)=∫X〈F,dμ〉 for F∈E.We prove a matricial version of the Richter–Tchakaloff theorem which states that each moment functional on E has a finitely atomic representing measure. It is shown that strictly positive linear functionals on E are moment functionals. For a moment functional Λ, we study the set of atoms W(Λ) and define two Carathéodory numbers of Λ. Further, we define and investigate the core set V(Λ) which generalizes the core variety from the scalar case to the matricial setting. A main result of the paper is the equality W(Λ)=V(Λ).

Description of the set of all possible masses at a fixed point of solutions of a truncated matricial Hamburger moment problem

We describe the set of all possible masses at a fixed point of the real axis in the context of the most general case of a truncated matricial (possibly degenerate) Hamburger moment problem. This set is a matricial interval, which we explicitly construct from the given data in the non-degenerate and degenerate cases simultaneously. In particular, we determine the maximum possible mass.

Indeterminate Hamburger moment problem: Entropy convergence

The indeterminate Hamburger moment problem is considered, jointly with all its real axis supported probability density functions. As a consequence of entropy functional concavity, out of such densities there is one which has largest entropy and that plays a fundamental role: we call it fhmax. It is proved that the approximate Maximum Entropy (MaxEnt) densities constrained by an increasing number of moments converge in entropy to fhmax where the value of its entropy can be finite or −∞.

A Note on the Matrix Moment Problem and its Relation with Matrix Biorthogonal Polynomials

A Note on the Matrix Moment Problem and its Relation with Matrix Biorthogonal Polynomials

Weyl Sets in a Non-degenerate Truncated Matricial Hausdorff Moment Problem

Given a point w in the upper half-plane Π+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi _{\\mathord {+}}$$\\end{document}, we describe the set of all possible values F(w) of transforms F(z):=∫[α,β](x-z)-1σ(dx)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(z)\\,{:=}\\,\\int _{[\\alpha ,\\beta ]}(x-z)^{-1}\\sigma (\ extrm{d}x)$$\\end{document}, z∈Π+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in \\Pi _{\\mathord {+}}$$\\end{document}, corresponding to solutions σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} to a (non-degenerate) truncated matricial Hausdorff moment problem. This set turns out to be the intersection of two matrix balls the parameters of which are explicitly constructed from the given data.

Shape, velocity, and exact controllability for the wave equation on a graph with cycle

Exact controllability is proved on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first applies a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated with exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails.

A Potapov-Type Approach to a Particular Truncated Stieltjes Moment Problem in the Case of an Odd Number of Prescribed Matricial Moments

The paper gives a parametrization of the solution set of a matricial Stieltjes-type truncated power moment problem in the non-degenerate and degenerate cases. The problem will be reformulated as an interpolation problem for a distinguished class of holomorphic matrix-valued functions. An essential role plays a solution of the corresponding system of Potapov’s fundamental matrix inequalities.

Multiplicity scaling of fragmentation function

The single-particle inclusive fragmentation function and the particle multiplicity are observables of fundamental importance in studying properties of quantum chromodynamics at colliders. It is well known that at high energies the multiplicity distribution satisfies Koba-Nielsen-Olesen (KNO) scaling in which all moments are proportional to powers of the mean multiplicity. We prove that, under weak assumptions, the leading dependence of the fragmentation function on multiplicity is itself a kind of KNO scaling in which all moments are inversely proportional to powers of the mean multiplicity. This scaling with multiplicity additionally accounts for the dominant dependence on collision energy in the fragmentation function. The proof relies crucially on properties of the fragmentation function conditioned on the total multiplicity and application of the Stieltjes moment problem. In the process, we construct a novel basis of the fragmentation function expressed as an overall exponential suppression times a series of Laguerre polynomials. We study this scaling of the fragmentation function in experimental electron-position collision data and observe that residual scale violations are significantly reduced. Published by the American Physical Society 2024

The realizability problem as a special case of the infinite-dimensional truncated moment problem

The realizability problem as a special case of the infinite-dimensional truncated moment problem

On the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem

On the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem

Sieving parton distribution function moments via the moment problem

We apply a classical mathematical problem, the moment problem, with its related mathematical achievements, to the study of the parton distribution function (PDF) in hadron physics, and propose a strategy to sieve the moments of the PDF by exploiting its properties such as continuity, unimodality, and symmetry. Through an error-inclusive sifting process, we refine three sets of PDF moments from Lattice QCD. This refinement significantly reduces the errors, particularly for higher order moments, and locates the peak of PDF simultaneously. As our strategy is universally applicable to PDF moments from any method, we strongly advocate its integration into all PDF moment calculations.

Indeterminate Stieltjes Moment Problem: Entropy Convergence

The aim of this paper is to consider the indeterminate Stieltjes moment problem together with all its probability density functions that have the positive real or the entire real axis as support. As a consequence of the concavity of the entropy function in both cases, there is one such density that has the largest entropy: we call it fhmax, the largest entropy density. We will prove that the Jaynes maximum entropy density (MaxEnt), constrained by an increasing number of integer moments, converges in entropy to the largest entropy density fhmax. Note that this kind of convergence implies convergence almost everywhere, with remarkable consequences in real applications in terms of the reliability of the results obtained by the MaxEnt approximation of the underlying unknown distribution, both for the determinate and the indeterminate case.