Hausdorff Moment Problem Research Articles

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.

The Born approximation in the three-dimensional Calderón problem Ⅱ: Numerical reconstruction in the radial case

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $ B_R \subset \mathbb{R}^3 $ of radius $ R $, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $ L^{\infty} $ conductivities, and present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $ \partial B_R $. We then investigate how increasing the radius $ R $ affects the quality of the Born approximation, and the existence of a scattering limit as $ R\to \infty $. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $ -\Delta +q $, and strong recovery of singularity results are observed in this case.

Beyond islands: a free probabilistic approach

We give a free probabilistic proposal to compute the fine-grained radiation entropy for an arbitrary bulk radiation state, in the context of the Penington-Shenker-Stanford-Yang (PSSY) model where the gravitational path integral can be implemented with full control. We observe that the replica trick gravitational path integral is combinatorially matching the free multiplicative convolution between the spectra of the gravitational sector and the matter sector respectively. The convolution formula computes the radiation entropy accurately even in cases when the island formula fails to apply. It also helps to justify this gravitational replica trick as a soluble Hausdorff moment problem. We then work out how the free convolution formula can be evaluated using free harmonic analysis, which also gives a new free probabilistic treatment of resolving the separable sample covariance matrix spectrum.The free convolution formula suggests that the quantum information encoded in competing quantum extremal surfaces can be modelled as free random variables in a finite von Neumann algebra. Using the close tie between free probability and random matrix theory, we show that the PSSY model can be described as a random matrix model that is essentially a generalization of Page’s model. It is then manifest that the island formula is only applicable when the convolution factorizes in regimes characterized by the one-shot entropies. We further show that the convolution formula can be reorganized to a generalized entropy formula in terms of the relative entropy.

Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution

We investigate the combinatorial sequences $A(M, n)$ introduced by W. G. Brown (1964) and W. T. Tutte (1980) appearing in enumeration of convex polyhedra. Their formula is $$A(M, n) = \frac{2 (2M+3)!}{(M+2)! M!}\,\frac{(4n+2M+1)!}{n! (3n + 2M + 3)!} $$ with $n, M =0, 1, 2, \ldots$, and we conceive it as Hausdorff moments, where $M$ is a parameter and $n$ enumerates the moments. We solve exactly the corresponding Hausdorff moment problem: $A(M, n) = \int_{0}^{R} x^{n} W_{M}(x) d x$ on the natural support $(0, R)$, $R = 4^{4}/3^{3}$, using the method of inverse Mellin transform. We provide explicitly the weight functions $W_{M}(x)$ in terms of the Meijer G-functions $G_{4, 4}^{4, 0}$, or equivalently, the generalized hypergeometric functions ${_{3}F_{2}}$ (for $M=0, 1$) and ${_{4}F_{3}}$ (for $M \geq 2$). For $M = 0, 1$, we prove that $W_{M}(x)$ are non-negative and normalizable, thus they are probability distributions. For $M \geq 2$, $W_{M}(x)$ are signed functions vanishing on the extremities of the support. By rephrasing this problem entirely in terms of Meijer G representations we reveal an integral relation which directly furnishes $W_M(x)$ based on ordinary generating function of $A(M, n)$ as an input. All the results are studied analytically as well as graphically.

On an Involution in the Class of Non-negative Hermitian Matrix Measures on a Compact Interval

We study a particular involution in the class {mathcal {M}^succcurlyeq _{q}({[alpha ,beta ]})} of all Non-negative Hermitian {qtimes q} measures on a compact interval {[alpha ,beta ]} of mathbb {R}. Several connections to the matricial version of the Hausdorff moment problem on {[alpha ,beta ]} are investigated in terms of {[alpha ,beta ]}-Stieltjes transforms as well as in the language of matricial Hausdorff moment sequences and associated matricial canonical moments.

A NOTE ON OPEN QUESTIONS ASKED TO ANALYSIS AND NUMERICS CONCERNING THE HAUSDORFF MOMENT PROBLEM

Abstract We address facts and open questions concerning the degree of ill-posedness of the composite Hausdorff moment problem aimed at the recovery of a function x ∈ L 2 (0, 1) from elements of the infinite dimensional sequence space ` 2 that characterize moments applied to the antiderivative of x. This degree, unknown by now, results from the decay rate of the singular values of the associated compact forward operator A, which is the composition of the compact simple integration operator mapping in L 2 (0, 1) and the non-compact Hausdorff moment operator B(H) mapping from L 2 (0, 1) to ` 2 . There is a seeming contradiction be- tween (a) numerical computations, which show (even for large n) an exponential decay of the singular values for n-dimensional matrices obtained by discretizing the operator A, and (b) a strongly limited smoothness of the well-known kernel k of the Hilbert-Schmidt operator A∗A. Fact (a) suggests severe ill-posedness of the infinite dimensional Hausdorff moment problem, whereas fact (b) lets us expect the opposite, because exponential ill-posedness oc- curs in common just for C∞-kernels k. We recall arguments for the possible occurrence of a polynomial decay of the singular values of A, even if the numerics seems to be against it, and discuss some issues in the numerical approximation of non-compact operators.

Hausdorff moment problem and nonlinear time optimality

A complete analytic solution for the time-optimal control problem for nonlinear control systems of the form ẋ1 = u, ẋj = ẋ1j−1, j = 2, …, n, is obtained for arbitrary n. In the paper we present the following surprising observation: this nonlinear optimality problem leads to a truncated Hausdorff moment problem, which gives analytic tools for finding the optimal time and optimal controls. Being homogeneous, the mentioned system approximates affine systems from a certain class in the sense of time optimality. Therefore, the obtained results can be used for solving the time-optimal control problem for systems from this class.

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan–Majda model and Taylor–Aris Dispersion

We study the long time behavior of an advection–diffusion equation with a random shear flow which depends on a stationary Ornstein–Uhlenbeck (OU) process in parallel-plate channels enforcing the no-flux boundary conditions. We derive a closed form formula for the long time asymptotics of the arbitrary N-point correlator using the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). In turn, appealing to the conclusion of the Hausdorff moment problem (Shohat and Tamarkin, 1943), we discover a diffusion equation with a random drift and deterministic enhanced diffusion possessing the exact same probability density function at long times. The strategy we presented is not only restricted to the parallel-plate channel domain. The same methods can derive effective equations for a straight channel with uniform arbitrary cross-section. Such equations enjoy many ergodic properties which immediately translate to ergodicity results for the original problem. In particular, we establish that the first two Aris moments using a single realization of the random field can be used to explicitly construct all ensemble averaged moments. Also, the first two ensemble averaged moments explicitly predict any long time centered Aris moment. Our formulae quantitatively depict the dependence of the deterministic effective diffusion on the interaction between spatial structure of flow and random temporal fluctuation. Further, this approximation provides many identities regarding the stationary OU process dependent time integral. We derive explicit formulae for the decaying passive scalar’s long time limiting probability density function (PDF) for different types of initial conditions (e.g. deterministic and random).

On the truncated Hausdorff moment problem under Sobolev regularity conditions

We study the problem of approximating the recovery of a probability distribution on the unit interval from its first k moments. As main result we obtain an upper bound on the L1 reconstruction error under the regularity assumption that the log-density function has square-integrable derivatives up to some natural order r>1. Our bound is of order O(k−r). A comparative study relates our findings to alternative conditions on the distributions.

A Schur–Nevanlinna Type Algorithm for the Truncated Matricial Hausdorff Moment Problem

The main goal of this paper is to achieve a parametrization of the solution set of the truncated matricial Hausdorff moment problem in the non-degenerate and degenerate situations. We treat the even and the odd cases simultaneously. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version, worked out in our former paper (Fritzsche et al.: Linear Algebra Appl 590:133–209. https://doi.org/10.1016/j.laa.2019.12.027, 2020), is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and discussion of the function-theoretic version is a central theme of this paper. This leads us to a complete description via Stieltjes transform of the solution set of the moment problem under consideration. Furthermore, we discuss special solutions in detail.

Rates of convergence in de Finetti’s representation theorem, and Hausdorff moment problem

Given a sequence $\{X_{n}\}_{n\geq 1}$ of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that $\frac{1}{n}\sum_{i=1}^{n}X_{i}\stackrel{a.s.}{\longrightarrow }Y$ for a suitable random variable $Y:\Omega \rightarrow [0,1]$ satisfying $\mathsf{P}[X_{1}=x_{1},\dots ,X_{n}=x_{n}|Y]=Y^{\sum_{i=1}^{n}x_{i}}(1-Y)^{n-\sum_{i=1}^{n}x_{i}}$. In this paper, we study the rate of convergence in law of $\frac{1}{n}\sum_{i=1}^{n}X_{i}$ to $Y$ under the Kolmogorov distance. After showing that a rate of the type of $1/n^{\alpha }$ can be obtained for any index $\alpha \in (0,1]$, we find a sufficient condition on the distribution of $Y$ for the achievement of the optimal rate of convergence, that is $1/n$. Besides extending and strengthening recent results under the weaker Wasserstein distance, our main result weakens the regularity hypotheses on $Y$ in the context of the Hausdorff moment problem.

Schur analysis of matricial Hausdorff moment sequences

We develop the algebraic instance of an algorithmic approach to the matricial Hausdorff moment problem on a compact interval [α,β] of the real axis. Our considerations are along the lines of the classical Schur algorithm and the treatment of the Hamburger moment problem on the real axis by Nevanlinna. More precisely, a transformation of matrix sequences is constructed, which transforms Hausdorff moment sequences into Hausdorff moment sequences reduced by 1 in length. It is shown that this transformation corresponds essentially to the left shift of the associated sequences of canonical moments. As an application, we show that a matricial version of the arcsine distribution can be used to characterize a certain centrality property of non-negative Hermitian measures on [α,β].

Hausdorff Moment Sequences Induced by Rational Functions

We study the Hausdorff moment problem for a class of sequences, namely $(r(n))_{n\in\mathbb Z_+},$ where $r$ is a rational function in the complex plane. We obtain a necessary condition for such sequence to be a Hausdorff moment sequence. We found an interesting connection between Hausdorff moment problem for this class of sequences with finite divided differences and convolution of complex exponential functions. We provide a sufficient condition on the zeros and poles of a rational function $r$ so that $(r(n))_{n\in\mathbb Z_+}$ is a Hausdorff moment sequence. G. Misra asked whether the module tensor product of a subnormal module with the Hardy module over the polynomial ring is again a subnormal module or not. Using our necessary condition we answer the question of G. Misra in negative. Finally, we obtain a characterization of all real polynomials $p$ of degree up to $4$ and a certain class of real polynomials of degree $5$ for which the sequence $(1/p(n))_{n\in\mathbb Z_+}$ is a Hausdorff moment sequence.

Hausdorff moment problem: Recovery of an unknown support for a probability density function

Abstract In this work the following problem is discussed: Unknown are two objects, a probability density (corresponding to a distribution, or to a probability measure) f and its support supp ⁡ { f } = [ L , U ] {\operatorname{supp}\{f\}=[L,U]} which is assumed to be bounded/compact. Available is the sequence of all integer order moments of f and the goal is to recover the support [ L , U ] {[L,U]} or both the support [ L , U ] {[L,U]} and the density f. This paper provides the sequences involving integer order moments that approximate the compact support of a probability measure. The proposed sequences are based on the ratios of subsequent moments and they converge faster than the previously suggested sequences representing the n-th roots of n-th moments. As a result, the approximation of a probability density function with unknown support is proposed. The convergence of newly defined approximations of the boundaries and the densities are demonstrated through various examples, including the case of noisy data.

On the numerical solution of the linear and nonlinear Poisson equations seen as bi-dimensional inverse moment problems

The numerical solution of the bi-dimensional nonlinear Poisson equations under Cauchy boundary conditions is considered. Using Green identity we show that this problem is equivalent to solve a bi-dimensional Fredholm integral equation of the first kind which can in turn be handled as a bi-dimensional generalized inverse moment problem. In the particular linear case the Helmholtz PDE is recovered and, within our scheme, the problem reduces to a bi-dimensional Hausdorff moment problem. In all the cases we find approximated solutions for the associated finite moment problems and bounds for the corresponding errors.

On Polynomial Maximum Entropy Method for Classical Moment Problem

AbstractThe maximum entropy method for the Hausdorff moment problem suffers from ill conditioning as it uses monomial basis {1,x,x2,...,xn}. The maximum entropy method for the Chebyshev moment probelm was studied to overcome this drawback in. In this paper we review and modify the maximum entropy method for the Hausdorff and Chebyshev moment problems studied in and present the maximum entropy method for the Legendre moment problem. We also give the algorithms of converting the Hausdorff moments into the Chebyshev and Lengendre moments, respectively, and utilizing the corresponding maximum entropy method.

Stochastic independence for probability MV-algebras

We propose a notion of stochastic independence for probability MV-algebras, addressing an open problem posed by Riečan and Mundici. Furthermore, we prove a representation theorem for probability MV-algebras and we get MV-algebraic versions of the Hölder inequality and the Hausdorff moment problem.

From the Potapov to the Krein–Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem

In their book The Markov Moment Problem and Extremal Problems, published in 1977, M. G. Krein and A. A. Nudel’man presented a complete solution of the truncated Hausdorff moment problem via orthogonal polynomials on a finite interval [a, b]. By using the Potapov schema the matrix version of this moment problem was studied by the author, Yu. M. Dyukarev, B. Fritzsche and B. Kirstein. In the present work, we obtain the matrix generalisation of the above-mentioned Krein–Nudel’man representation. We also obtain explicit relations between four families of orthogonal matrix polynomials on [a, b] and their second kind polynomials, which are associated with the matrix version of the truncated Hausdorff moment problem.

Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators, part II: the use of mathematical fluctuation theory

The first part of these two companion papers has been devoted to the extension of Hausdorff moment problem to the sequences over integrals of Kronecker powers of an appropriate vector under a generating function in the kernel. The relations between this generating function and weight function properties have been investigated over there in a quite detailed manner. This second companion paper focuses on the utilization of the “mathematical fluctuation theory” amenities in the construction of approximations to the solutions of the expectation value dynamics of the quantum dynamical systems. The fluctuation freee approximation matching with the classical mechanical behaviour is followed by the first and then the second order fluctuation approximations. Beside the well known “Energy Conservation Law”s counterparts in these approximations of quantum expectation value dynamics are also presented.

On exchangeability and Hausdorff moment problems over k-dimensional simplexes

Two classical problems, the Hausdorff moment problem and de Finetti's representation theorem for exchangeable random variables, are considered. We generalize these problems to a g-tuple of k-dimensional simplexes and an infinite sequence of g-fold partially exchangeable random variables in {0,1,...,k}, respectively. The equivalence between them is then formalized and proven.