Classical Moment Problem Research Articles

Higher correlations of divisor sums related to primes III: small gaps between primes

We use divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η > 0, a positive proportion of consecutive primes are within 4 + η times the average spacing between primes. Authors’ note. This paper was written in 2004, prior to the solution, in [8], of the problem considered here. In [8] it is shown that Δ = 0. While the main result in Theorem 1 has now been superseded, we believe the method used here is both of interest and future utility in other applications. In particular, the work of Green and Tao [12] on arithmetic progressions of primes makes use of Proposition 1 of this paper.

Orthogonal polynomials for weights close to indeterminacy

We obtain estimates for Christoffel functions and orthogonal polynomials for even weights W : R → [ 0 , ∞ ) that are ‘close’ to indeterminate weights. Our main example is exp - x log x β , with β real, possibly modified near 0, but our results also apply to exp - x α log x β , α < 1 . These types of weights exhibit interesting properties largely because they are either indeterminate or are close to the border between determinacy and indeterminacy in the classical moment problem.

What non-linear methods offered to linear problems? The Fokas transform method

In 1750 D’ Alembert demonstrated how a linear partial differential equation can be solved via separation of variables, a method that decomposes a PDE into a set of ODEs. This method was the basis for the development of many branches of contemporary analysis, from function spaces to spectral analysis of operators and the theory of special functions. A condition for the method of separation of variables to work is the existence of a coordinate system that fits the boundary of the fundamental domain and at the same time it separates the PDE. It is remarkable that two and a half centuries later a generalization is introduced that has its origin in the analysis of non-linear integrable equations. In the present work, this promising new transform method is outlined and applied to particular boundary value problems. A crucial part of the method is the introduction of a global relation which, if properly used, can provide the missing boundary data in a very elegant and effective way. We show how this can be used to generate separable solutions of partial differential equations even when no system, that fits the geometry of the fundamental domain, is available. This is shown for the case of the Dirichlet problem for the modified Helmholtz equation in the interior of an equilateral triangle. Furthermore, the connection of the Fokas method to the classical moment problem is investigated. It is shown that, in this case, the global relation is decomposed into a sequence of global relations, directly associated with the Fourier coefficients of the Dirichlet and Neumann boundary values.

Generalized coherent states for oscillators connected with Meixner and Meixner—Pollaczek polynomials

The authors continue to study generalized coherent states for oscillator-like systems connected with a given family of orthogonal polynomials. In this work, we consider oscillators connected with Meixner and Meixner— Pollaczek polynomials and define generalized coherent states for these oscillators. A completeness condition for these states is proved by solution of a related classical moment problem. The results are compared with the other authors ones. In particular, we show that the Hamiltonian of the relativistic model of a linear harmonic oscillator can be treated as the linearization of a quadratic Hamiltonian, which arises naturally in our formalism. Bibliography: 56 titles.

The total π-electron energy as a problem of moments: application of the Backus–Gilbert method

The total π-electron energy problem can be formulated as a classical problem of moments. This observation allows us to apply general methodologies developed in the field of moment’s theory to solve the total π-electron energy problem. In the present article, we apply the Backus–Gilbert method to obtain analytical expressions for the total π-electron energy in terms of its spectral moments.

Stieltjes classes for moment-indeterminate probability distributions

LetFbe a probability distribution function with densityf. We assume that (a) F has finite moments of any integer positive order and (b) the classical problem of moments forFhas a nonunique solution (Fis M-indeterminate). Our goal is to describe a, wherehis a ‘small' perturbation function. Such a classSconsists of different distributionsFε(fεis the density ofFε) all sharing the same moments as those ofF, thus illustrating the nonuniqueness ofF, and of anyFε,in terms of the moments. Power transformations of distributions such as the normal, log-normal and exponential are considered and for them Stieltjes classes written explicitly. We define a characteristic ofScalled anindex of dissimilarityand calculate its value in some cases. A new Stieltjes class involving a power of the normal distribution is presented. An open question about the inverse Gaussian distribution is formulated. Related topics are briefly discussed.

Stieltjes classes for moment-indeterminate probability distributions

LetFbe a probability distribution function with densityf. We assume that (a) F has finite moments of any integer positive order and (b) the classical problem of moments forFhas a nonunique solution (Fis M-indeterminate). Our goal is to describe a, wherehis a ‘small' perturbation function. Such a classSconsists of different distributionsFε(fεis the density ofFε) all sharing the same moments as those ofF, thus illustrating the nonuniqueness ofF, and of anyFε,in terms of the moments. Power transformations of distributions such as the normal, log-normal and exponential are considered and for them Stieltjes classes written explicitly. We define a characteristic ofScalled anindex of dissimilarityand calculate its value in some cases. A new Stieltjes class involving a power of the normal distribution is presented. An open question about the inverse Gaussian distribution is formulated. Related topics are briefly discussed.

The Generalized Moment Problem Associated with Correlation Measures

The classical power moment problem can be viewed as a theory of spectral representations of a positive functional on some classical commutative algebra with involution. We generalize this approach to the case in which the algebra is a special commutative algebra of functions on the space of multiple finite configurations. If the above-mentioned functional is generated by a measure on the space of finite ordinary configurations, then this measure is the correlation measure for a measure on the space of infinite configurations. The positiveness of the functional gives conditions for the measure to be a correlation measure.

Nevanlinna Matrices for the Strong Hamburger Moment Problem

Let {c n }∞n=-∞ be a doubly infinite sequence of real numbers. The strong Hamburger moment problem consists of finding positive measures σ on R such that c n = ∫∞ - ∞ t n dσ(t) for n = 0, ±1, ±2,.... The problem is indeterminate if there is more than one solution. For an indeterminate problem there is a one-to-one correspondence between all Pick functions φ and all solutions σ of the moment problem, expressed by ∫∞ - ∞t) -1 dσ(t) = [α(z)φ(z) - γ(z)][β(z)φ(z) - δ(z)] -1 . The functions α,β,γ,δ are holomorphic in the complex plane outside the origin. The purpose of this paper is to study growth properties of these functions a, β,γ,δ, analogous to properties of corresponding entire functions connected with the classical Hamburger moment problem.

Moment problems and the causal set approach to quantum gravity

We study a collection of discrete Markov chains related to the causal set approach to modeling discrete theories of quantum gravity. The transition probabilities of these chains satisfy a general covariance principle, a causality principle, and a renormalizability condition. The corresponding dynamics are completely determined by a sequence of non-negative real coupling constants. Using techniques related to the classical moment problem, we give a complete description of any such sequence of coupling constants. We prove a representation theorem: every discrete theory of quantum gravity arising from causal set dynamics satisfying covariance, causality, and renormalizability corresponds to a unique probability distribution function on the non-negative real numbers, with the coupling constants defining the theory given by the moments of the distribution.

Continued fractions and integrable systems

Stieltjes’ solution of the classical moment problem is the forerunner of inverse spectral theory. In the modern theory of completely integrable systems, the method of inverse scattering is used to obtain explicit solutions of a class of Hamiltonian systems which arise as isospectral deformations of a linear operator. In this lecture we explain how Stieltjes's formulas are used to obtain explicit solutions to a number of completely integrable systems, including discrete reductions of some nonlinear partial differential equations, and the isospectral deformations of Jacobi matrices a.k.a. Toda flows.

On some issues related to the moment problem for the band matrices with operator elements

The connection between the classical moment problem and the spectral theory of second order difference operators (or Jacobi matrices) is a thoroughly studied topic. Here we examine a similar connection in the case of the second order operator replaced by an operator generated by an infinite band matrix with operator elements. For such operators, we obtain an analog of the Stone theorem and consider the inverse spectral problem which amounts to restoring the operator from the moment sequence of its Weyl matrix. We establish the solvability criterion for such problems, find the conditions ensuring that the elements of the moment sequence admit an integral representation with respect to an operator valued measure and discuss an algorithm for the recovery of the operator. We also indicate a connection between the inverse problem method and the Hermite–Padé approximations.

Some generalizations of the classical moment problem

The article is devoted to two generalizations of the classical power moment problem, namely: 1) instead of representing the moment sequence by λn, a representation by polynomialsPn(λ), ℝ1, connected with a Jacobi matrix, appears; 2) in the representation, instead of λn, the expression λ⊗n figures, where λ is a real generalized function (i.e., we investigate some infinite-dimensional moment problem).

The uniqueness question in the multidimensional moment problem with applications to phase space observables

The theory of holomorphic functions of several complex variables is applied in proving a multidimensional variant of a theorem involving an exponential boundedness criterion for the classical moment problem. A theorem of Petersen concerning the relation between the multidimensional and one-dimensional moment problems is extended for half-lines and compact subsets of the real line. These results are used to solve the moment problem for the quantum phase space observables generated by the number states.

On a Special Caratheodory Subclass and the Set of Solvability of the Markov Trigonometric Moment Problem with Periodic Gaps

The classical Markov trigonometric moment problem is considered in the case when the moment sequence has periodically repeating gaps. The suggested approach represent a development of the N. I. Akhieser’s and M.G. Krein’s idea on the relation between the trigonometric moment problem and the Caratheodory coefficients problem. We associate with the gaps distribution law M a certain subclass C(M ) of the Caratheodory functions class C. The multiplicative representation of the class C(M ) is obtained. On this basis the complete description of the solvability set of the Markov trigonometric moment problem with gaps as well as a method of searching of its canonical solutions are given.

A rational Stieltjes moment problem

Let { α k } be a sequence of points on the real axis, and let L denote the linear span of the set 1 ω 0 , 1 ω 1 ,…, 1 ω n ,… , where ω 0=1,ω n=(z−α 1)(z−α 2)⋯(z−α n) for n⩾1 . A measure μ with support in an interval [ γ,∞) and with respect to which all functions in L· L are integrable induces an inner product in L and hence orthogonal rational functions { ϕ n } and associated functions { σ n } corresponding to the sequence {1/ ω n }. Assuming certain constellations of the points { α k }, it is shown that the sequences − σ 2m(z) ϕ 2m(z) and − σ 2m+1(z) ϕ 2m+1(z) are monotonic on a certain real interval ( α, β) and convergent in the complex plane outside the interval [ γ,∞) to functions F( z, μ (0)) and F( z, μ (∞)), where F( z, μ) denotes the Stieltjes transform ∫ −∞ ∞d μ( t)/( t− z) of a measure μ. Furthermore for every μ giving rise to the same integral values for functions in L· L , the inequality F( x, μ (0))⩽ F( x, μ)⩽ F( x, μ (∞)) holds for x∈( α, β). These results are essentially generalizations of results concerning the strong (or two-point) Stieltjes moment problem, and are also similar to results concerning the classical Stieltjes moment problem.

Remarks on converse Carleman and Krein criteria for the classical moment problem

AbstractThe key theme is converse forms of criteria for deciding determinateness in the classical moment problem. A method of proof due to Koosis is streamlined and generalized giving a convexity condition under which moments satisfying implies that c a positive constant. A contrapositive version is proved under a rapid variation condition on f (x), generalizing a result of Lin. These results are used to obtain converses of the Stieltjes versions of the Carleman and Krein criteria. Hamburger versions are obtained which relax the symmetry assumption of Koosis and Lin, respectively. A sufficient condition for Stieltjes determinateness of a discrete law is given in terms of its mass function. These criteria are illustrated through several examples.

Backus-Gilbert algorithm for the Cauchy problem ofthe Laplace equation

In this paper, the highly ill posed Cauchy problem for the Laplaceequation is transformed to a classical moment problem whosenumerical approximation can be achieved. Proofs on itsconvergence and stability estimatesare given based on the Backus-Gilbert algorithm. For numericalverification, several examples which include random noise inthe initial Cauchy data are presented.

From Finite Covariance Windows to Modeling Filters: A Convex Optimization Approach

The trigonometric moment problem is a classical moment problem with numerous applications in mathematics, physics, and engineering. The rational covariance extension problem is a constrained version of this problem, with the constraints arising from the physical realizability of the corresponding solutions. Although the maximum entropy method gives one well-known solution, in several applications a wider class of solutions is desired. In a seminal paper, Georgiou derived an existence result for a broad class of models. In this paper, we review the history of this problem, going back to Carath{éodory, as well as applications to stochastic systems and signal processing. In particular, we present a convex optimization problem for solving the rational covariance extension problem with degree constraint. Given a partial covariance sequence and the desired zeros of the shaping filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this way we obtain an algorithm for solving the covariance extension problem, as well as a constructive proof of Georgiou's existence result and his conjecture, a generalized version of which we have recently resolved using geometric methods. We also survey recent related results on constrained Nevanlinna--Pick interpolation in the context of a variational formulation of the general moment problem.

Multipeakons and the Classical Moment Problem

Classical results of Stieltjes are used to obtain explicit formulas for the peakon–antipeakon solutions of the Camassa–Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon–antipeakon pairs, and the details of the collisions are analyzed using results from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly.