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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. [ABSTRACT FROM AUTHOR]

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = il£) with L (or £) and U ( or il) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J^' = UL (or = £il), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations. [ABSTRACT FROM AUTHOR]

The occasion for this survey article was the 70th birthday of Jan Stochel, professor at Jagiellonian University, former head of the Chair of Functional Analysis and a prominent member of the Kraków school of operator theory. In the course of his mathematical career, he has dealt, among other things, with various aspects of functional analysis, single and multivariable operator theory, the theory of moments, the theory of orthogonal polynomials, the theory of reproducing kernel Hilbert spaces, and mathematical aspects of quantum mechanics. [ABSTRACT FROM AUTHOR]

We use the moment method in order to study the uniform boundary controllability of a semidiscrete 1-d wave equation, when a lower rate numerical vanishing viscosity term is added. The high frequency spurious oscillations introduced by the classical method of space discrete numerical schemes lead to nonuniform controllability properties, which are dumped out using an additional vanishing viscosity term. Our extra numerical viscosity is weaker than the one used in [30] , but enough in order to treat high frequency numerical spurious oscillations. Hence, we are able to prove the convergence of the sequence of discrete controls to a control of the continuous wave equation, when the mesh size tends to zero. [ABSTRACT FROM AUTHOR]

In this article we evaluate the cost of the finite dimensional and the approximate controllability of a one dimensional anomalous diffusion equation which is not null-controllable. The finite dimensional controllability property consists in driving to zero a finite number of frequencies of the corresponding solutions. To evaluate its cost, we construct an explicit biorthogonal sequence to a finite part of a family of real exponential functions. Since the exponents of these functions satisfy the so-called Müntz density condition, there is no biorthogonal sequence to the entire family which shows that the equation is not even spectrally controllable. For the approximate controllability problem, we combine the idea of finite dimensional controllability studied before and the intrinsic dissipative nature of the system. These arguments allow to show that the controlled solution can be made arbitrarily small after sufficiently large time and provide estimates for the corresponding approximate controls. [ABSTRACT FROM AUTHOR]

The occasion for this survey article was the 70th birthday of Jan Stochel, professor at Jagiellonian University, former head of the Chair of Functional Analysis and a prominent member of the Kraków school of operator theory. In the course of his mathematical career, he has dealt, among other things, with various aspects of functional analysis, single and multivariable operator theory, the theory of moments, the theory of orthogonal polynomials, the theory of reproducing kernel Hilbert spaces, and mathematical aspects of quantum mechanics.

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.

The properties of the unique nontrivial analytic solution, defined implicitly by a functional equation, are pointed out. This work provides local estimations and global inequalities for the involved solution. The corresponding operatorial equation is studied as well. The second part of the paper is devoted to the full classical moment problem, which is an inverse problem. Two constraints are imposed on the solution. One of them requires the solution to be dominated by a concrete convex operator defined on the positive cone of the domain space. A one-dimensional operator is valued, and a multidimensional scalar moment problem is solved. In both cases, the existence and the uniqueness of the solution are proved. The general idea of the paper is to provide detailed information on solutions which are not expressible in terms of elementary functions. [ABSTRACT FROM AUTHOR]

The general goal of this work is to obtain upper and lower bounds for the $ L^2 $-norm of biorthogonal families to complex exponential functions associated to sequences $ \left\{{ \Lambda_k}\right\}_{k \ge 1} \subset {\mathbb{{C}}} $ which satisfy appropriate assumptions but without imposing a gap condition on the elements of the sequence. As a consequence, we also present new results on the cost of the boundary null controllability of two parabolic systems at time $ T > 0 $: a phase-field system and a parabolic system whose generator has eigenvalues that accumulate. In the latter case, the behavior of the control cost when $ T $ goes to zero depends strongly on the accumulation parameter of the eigenvalue sequence. [ABSTRACT FROM AUTHOR]

It is well-known that any L 2 -martingale with respect to a Brownian filtration is represented by a stochastic integral with respect to the Brownian motion. The theorem can be proven based on the fact that linear combinations of exponential martingales (of a specific type) are dense in the mentioned set. In this paper, the necessary and sufficient conditions for expressing martingales as true identities rather than approximations are considered, which turns out to be an infinite dimensional moment problem. A typical moment problem is given as follows: for real sequences (μ i) i = 0 ∞ , find the necessary and sufficient conditions for the existence of a distribution whose support is a subset of [ 0 , ∞) and the i -th moments is μ i . This is a fundamental problem in probability theory or integral theory that was first proposed around 1894, but it is still being studied as of 2023. In this paper, we point out that this problem is related to the above problem through chaos expansion, and give a proof using a version of the moment problem. [ABSTRACT FROM AUTHOR]

We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the linear equation of parabolic type and the system controlled by impulsive external influences. Our optimal control problem deals with finding a control parameter belonging to the class of admissible controls that provides the desired temperature distribution in a finite time with minimal energy consumption (energy consumption is described by the quadratic functional). Previous works dedicated to optimal impulse control problems have mostly used the Pontryagin's maximum principle. However, from a practical point of view, this approach does not lead to satisfactory results. This is due to the fact that the corresponding boundary value problems in this case have no solution in a traditional class of absolutely continuous trajectories. In this work, we propose a method based on the moment relations. We seek for the approximate solution of the corresponding boundary value problem in the form of finite Fourier sum and state our optimal control problem in a finite-dimensional phase space. As a result, we obtain an optimal impulse control problem in a finite-dimensional function space. Taking into account the given condition for a finite time, we reduce the obtained problem to the L-problem of moments. Thus, the problem of finding a control parameter is reduced to the solution of the system of Fredholm integral equations of the first kind, with the norm of the sought solution not exceeding a given number. By Levi's theorem, every element of Hilbert space can be represented by the sum of the elements of two orthogonal subspaces. This assertion makes it possible to find control parameters in analytical form. We also establish the convergence of the chosen approximation. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the system of left translates {L(2k,l,m)g: k, l,m ∈ Z}, g ∈ L²(H), to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function gλ. Here, H denotes the Heisenberg group and gλ the inverse Fourier transform of g with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates {L(2k,l,m)g: k, l, m ∈ Z} on H. This result is also illustrated with an example. [ABSTRACT FROM AUTHOR]

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = 𝔘𝔏) with L (or 𝔏) and U (or 𝔘) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J(p) = UL (or J(d) = 𝔏𝔘), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations.

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.

In this paper we consider the following moment problem: find a positive Borel measure μ on ℂ subject to conditions ∫ zn dμ = sn, n∈ℤ+, where sn are prescribed complex numbers (moments). This moment problem may be viewed (informally) as an extension of the Stieltjes and Hamburger moment problems to the complex plane. A criterion for the moment problem for the existence of a compactly supported solution is given. In particular, such moment problems appear naturally in the domain of complex Jacobi matrices. For every bounded complex Jacobi matrix its associated functional S has the following integral representation: S(p) = ∫ℂ p(z) dμ, with a positive Borel measure μ in the complex plane. An interrelation of the associated to the complex Jacobi matrix operator A0, acting in l2 on finitely supported vectors, and the multiplication by z operator in L2μ is discussed.

The paper deals with the controllability properties of the Kawahara equation posed on a periodic domain. We show that the equation is exactly controllable by means of a control depending only on time and acting on the system through a given shape function in space. Firstly, the exact controllability property is established for the linearized system through a Fourier expansion of solutions and the analysis of a biorthogonal sequence to a family of complex exponential functions. Finally, the local controllability of the full system is derived by combining the analysis of the linearized system, a fixed point argument and some Bourgain smoothing properties of the Kawahara equation on a periodic domain. [ABSTRACT FROM AUTHOR]

In this paper, the decoding problem for multiple Bose–Chaudhuri–Hocquenghem (BCH)‐Goppa codes over the same finite field is investigated. A new iterative decoding algorithm is proposed based on the quotient difference (qd)‐type discrete hungry Lotka–Volterra equation over finite fields. Compared with certain existing algorithms, the proposed algorithm manifests its advantage in computational complexity. A few of examples are presented to demonstrate its efficiency. [ABSTRACT FROM AUTHOR]

The purpose of this work is to provide applications of real, complex, and functional analysis to moment, interpolation, functional equations, and optimization problems. Firstly, the existence of the unique solution for a two-dimensional full Markov moment problem is characterized on the upper half-plane. The issue of the unknown form of nonnegative polynomials on R × R + in terms of sums of squares is solved using polynomial approximation by special nonnegative polynomials, which are expressible in terms of sums of squares. The main new element is the proof of Theorem 1, based only on measure theory and on a previous approximation-type result. Secondly, the previous construction of a polynomial solution is completed for an interpolation problem with a finite number of moment conditions, pointing out a method of determining the coefficients of the solution in terms of the given moments. Here, one uses methods of symmetric matrix theory. Thirdly, a functional equation having nontrivial solution (defined implicitly) and a consequence are discussed. Inequalities, the implicit function theorem, and elements of holomorphic functions theory are applied. Fourthly, the constrained optimization of the modulus of some elementary functions of one complex variable is studied. The primary aim of this work is to point out the importance of symmetry in the areas mentioned above. [ABSTRACT FROM AUTHOR]

In this paper, we aim to provide new and simple proof for the determinant characterization of moment sequences with discrete measures by investing in generalized Fibonacci sequences. [ABSTRACT FROM AUTHOR]

We propose an algorithm which is based on the the technique introduced in [23]. The aim of the algorithm is to study, in a simple way, the approximation of the controls for a class of hyperbolic problems. It is well-known that, the finite-difference semi-discrete scheme for the approximation of controls can leads to high frequency numerical spurious oscillations which gives a loss of the uniform (with respect to the mesh-size) controllability property of the semi-discrete model. It is also known that an appropriate filtration of the high eigenfrequencies of the discrete initial data enable us to restore the uniform controllability property of the whole solution. But, the methods used to prove such results are very constructive and use difficult and fine computations. As an example, which proves the effectiveness of our algorithm, we consider the case of the semidiscrete one dimensional wave equation. In this particular case, we are able to prove the uniform controllability, where the initial data are filtered in a range which contains as many modes as possibles, taking into account previous results obtained in literature (see [18]). [ABSTRACT FROM AUTHOR]

Starting from measured data, we develop a method to compute the finestructure of the spectrum of the Koopman operator with rigorous convergenceguarantees. The method is based on the observation that, in themeasure-preserving ergodic setting, the moments of the spectral measureassociated to a given observable are computable from a single trajectory ofthis observable. Having finitely many moments available, we use the classicalChristoffel-Darboux kernel to separate the atomic and absolutely continuousparts of the spectrum, supported by convergence guarantees as the number ofmoments tends to infinity. In addition, we propose a technique to detect thesingular continuous part of the spectrum as well as two methods to approximatethe spectral measure with guaranteed convergence in the weak topology,irrespective of whether the singular continuous part is present or not. Theproposed method is simple to implement and readily applicable to large-scalesystems since the computational complexity is dominated by inverting an$N\times N$ Hermitian positive-definite Toeplitz matrix, where $N$ is thenumber of moments, for which efficient and numerically stable algorithms exist;in particular, the complexity of the approach is independent of the dimensionof the underlying state-space. We also show how to compute, from measured data,the spectral projection on a given segment of the unit circle, allowing us toobtain a finite-dimensional approximation of the operator that explicitly takesinto account the point and continuous parts of the spectrum. Finally, wedescribe a relationship between the proposed method and the so-called HankelDynamic Mode Decomposition, providing new insights into the behavior of theeigenvalues of the Hankel DMD operator. A number of numerical examplesillustrate the approach, including a study of the spectrum of the lid-driventwo-dimensional cavity flow.

A new method to prove exact controllability for the wave equation is demonstrated and discussed on several examples. The method of proof first uses 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 to exact controllability.

In this paper, we study the quadratic tensor eigenvalue complementarity problem (QTEiCP). By a randomization process, the quadratic com-plementarity(QC) eigenvalues are classified into two cases. For each case, the QTEiCP is formulated as an equivalent generalized moment problem. The QC eigenvectors can be computed in order. Each of them can be solved by a sequence of semidefinite relaxations. We prove that such a sequence converges in finitely many steps for generic tensors. [ABSTRACT FROM AUTHOR]

We study a limiting case of the Askey–Wilson polynomials when one of the parameters goes to infinity, namely continuous dual q-Hahn polynomials when q>1. Solutions to the associated indeterminate moment problem by general theory are found and an orthogonality relation is established. [ABSTRACT FROM AUTHOR]

In this paper we consider the following moment problem: find a positive Borel measure μ on C subject to conditions ∫ zndμ = sn, n ∈ Z+, where sn are prescribed complex numbers (moments). This moment problem may be viewed (informally) as an extension of the Stieltjes and Hamburger moment problems to the complex plane. A criterion for the moment problem for the existence of a compactly supported solution is given. In particular, such moment problems appear naturally in the domain of complex Jacobi matrices. For every bounded complex Jacobi matrix its associated functional S has the following integral representation: S(p) = ∫ C p(z)dμ, with a positive Borel measure μ in the complex plane. An interrelation of the associated to the complex Jacobi matrix operator A0, acting in l² on finitely supported vectors, and the multiplication by z operator in L² μ is discussed. [ABSTRACT FROM AUTHOR]

We study the relation between certain non-degenerate lower Hessenberg infinite matrices G and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known Favard theorem for Sobolev orthogonality. We characterize the structure of the matrix G and the associated matrix of formal moments M G in terms of certain matrix operators. [ABSTRACT FROM AUTHOR]

We study the infinite Jacobi block matrices under the discrete Miura-type transformations which relate matrix Volterra and Toda lattice systems to each other and the situations when the deficiency indices of the corresponding operators are the same. A special attention is paid to the completely indeterminate case (i.e., then the deficiency indices of the corresponding block Jacobi operators are maximal). It is shown that there exists a Miura transformation which retains the complete indeterminacy of Jacobi block matrices appearing in the Lax representation for such systems, namely, if the Lax matrix of Volterra system is completely indeterminate, then so is the Lax matrix of the corresponding Toda system, and vice versa. We consider an implication of the obtained results to the study of matrix orthogonal polynomials as well as to the analysis of self-adjointness of scalar Jacobi operators.

This paper develops various optimization techniques to estimate probability of events where the optimal value of a convex program, satisfying certain structural assumptions, exceeds a given threshold. First, we relate the search of affine/polynomial policies for the robust counterpart to existing relaxation hierarchies in MINLP (Lasserre in Proceedings of the international congress of mathematicians (ICM 2018), 2019; Sherali and Adams in A reformulation–linearization technique for solving discrete and continuous nonconvex problems, Springer, Berlin). Second, we leverage recent advances in Dworkin et al. (in: Kaski, Corander (eds) Proceedings of the seventeenth international conference on artificial intelligence and statistics, Proceedings of machine learning research, PMLR, Reykjavik, 2014), Gawrychowski et al. (in: ICALP, LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018) and Rizzi and Tomescu (Inf Comput 267:135–144, 2019) to develop techniques to approximately compute the probability binary random variables from Bernoulli distributions belong to a specially-structured union of sets. Third, we use convexification, robust counterpart, and chance-constrained optimization techniques to cover the event set of interest with such set unions. Fourth, we apply our techniques to the network reliability problem, which quantifies the probability of failure scenarios that cause network utilization to exceed one. Finally, we provide preliminary computational evaluation of our techniques on test instances for network reliability. [ABSTRACT FROM AUTHOR]

We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions. [ABSTRACT FROM AUTHOR]

This paper is devoted to analyzing the controllability to rest of the Gurtin-Pipkin model, which is a class of differential equations with memory terms. The goal is not only to derive the state to vanish at some time but also to require the memory term to vanish at the same time, ensuring that the controlled system is controllable to rest. In order to get rid of the influence of memory, the controllability result is obtained by means of the Fourier type approach and the moment theory. [ABSTRACT FROM AUTHOR]

The purpose of this work is to provide applications of real, complex, and functional analysis to moment, interpolation, functional equations, and optimization problems. Firstly, the existence of the unique solution for a two-dimensional full Markov moment problem is characterized on the upper half-plane. The issue of the unknown form of nonnegative polynomials on R×R+ in terms of sums of squares is solved using polynomial approximation by special nonnegative polynomials, which are expressible in terms of sums of squares. The main new element is the proof of Theorem 1, based only on measure theory and on a previous approximation-type result. Secondly, the previous construction of a polynomial solution is completed for an interpolation problem with a finite number of moment conditions, pointing out a method of determining the coefficients of the solution in terms of the given moments. Here, one uses methods of symmetric matrix theory. Thirdly, a functional equation having nontrivial solution (defined implicitly) and a consequence are discussed. Inequalities, the implicit function theorem, and elements of holomorphic functions theory are applied. Fourthly, the constrained optimization of the modulus of some elementary functions of one complex variable is studied. The primary aim of this work is to point out the importance of symmetry in the areas mentioned above.

We investigate a class of moment problems, namely recovering a measure supported on the graph of a function from partial knowledge of its moments, as, for instance, in some problems of optimal transport or density estimation. We show that the sole knowledge of first degree moments of the function, namely linear measurements, is sufficient to obtain asymptotically all the other moments by solving a hierarchy of semidefinite relaxations (viewed as moment matrix completion problems) with a specific sparsity-inducing criterion related to a weighted ℓ 1 -norm of the moment sequence of the measure. The resulting sequence of optimal solutions converges to the whole moment sequence of the measure which is shown to be the unique optimal solution of a certain infinite-dimensional linear optimization problem (LP). Then one may recover the function by a recent extraction algorithm based on the Christoffel–Darboux kernel associated with the measure. Finally, the support of such a measure supported on a graph is a meager, very thin (hence sparse) set. Therefore, the LP on measures with this sparsity-inducing criterion can be interpreted as an analogue for infinite-dimensional signals of the LP in super-resolution for (sparse) atomic signals. [ABSTRACT FROM AUTHOR]

We study the tight upper bound of the transient mean waiting time in the classical GI/GI/1 queue over candidate interarrival-time distributions with finite support, given the first two moments of the interarrival time and the full service-time distribution. We formulate the problem as a non-convex nonlinear program. We derive the gradient of the transient mean waiting time and then show that a stationary point of the optimization can be characterized by a linear program. We develop and apply a stochastic variant of the Frank and Wolfe (Naval Res Logist Q 3:95–110, 1956) algorithm to find a stationary point for any given service-time distribution. We also establish necessary conditions and sufficient conditions for stationary points to be three-point distributions or special two-point distributions. We illustrate by applying simulation together with optimization to analyze several examples. [ABSTRACT FROM AUTHOR]

The least upper bound for the probability that a random vector with fixed mean and covariance will be outside the ball is found. This probability bound is determined by solving a scalar equation and, in the case of identity covariance matrix, is given by an analytical expression, which is a multivariate generalization of the Selberg bound. It is shown that at low probability levels, it is more typical when the bound is given by the new expression if compared with the case when it coincides with the right-hand side of the well-known Markov inequality. The obtained result is applied to solving the problem of hypothesis testing by using an alternative with uncertain distribution. [ABSTRACT FROM AUTHOR]

We deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra A be represented as an integral with respect to a Radon measure on the character space X(A) of A equipped with the Borel σ -algebra generated by the weak topology? We approach this problem by constructing X(A) as a projective limit of the character spaces of all finitely generated unital subalgebras of A. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on A with respect to a measure on the cylinder σ -algebra on X(A) (resp. a Radon measure on the Borel σ -algebra on X(A)) provided that for any finitely generated unital subalgebra of A the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on A. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a "partially Archimedean" quadratic module of A. Our results in particular apply to the case when A is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing a unified setting which enables comparisons between some recent results for these instances of the moment problem. [ABSTRACT FROM AUTHOR]

We introduce a framework for generating samples of a distribution given a finite number of its moments, targeted at particle-based solutions of kinetic equations and rarefied gas flow simulations. Our model, referred to as the Wasserstein-Entropy distribution (WE), couples a physically-motivated Wasserstein penalty term to the traditional maximum-entropy distribution (MED) function, which serves to regularize the latter. The penalty term becomes negligible near the local equilibrium, reducing the proposed model to the MED, known to reproduce the hydrodynamic limit. However, in contrast to the standard MED, the proposed WE closure can cover the entire physically realizable moment space, including the so-called Junk line. We also propose an efficient Monte Carlo algorithm for generating samples of the unknown distribution which is expected to outperform traditional non-linear optimization approaches used to solve the MED problem. Numerical tests demonstrate that given moments up to the heat flux—that is, information equivalent to that contained in the Chapman-Enskog distribution—the proposed methodology provides a reliable closure in the collision-dominated and early transition regimes. Applications to greater rarefaction demand information from higher-order moments, which can be incorporated within the proposed closure. • Generate samples from distribution defined by small number of its moments. • Solution based on entropic optimal transport. • Resulting distribution is well defined for all realizable moments. • Samples generated using efficient Monte Carlo algorithm. • Validated using DSMC simulations of rarefied gas hydrodynamics. [ABSTRACT FROM AUTHOR]

We establish a criterion for the existence of a representing Radon measure for linear functionals defined on a unital commutative real algebra A , which we assume to be generated by a vector space V endowed with a Hilbertian seminorm q. Such a general criterion provides representing measures with support contained in the space of characters of A whose restrictions to V are q −continuous. This allows us in turn to prove existence results for the case when V is endowed with a nuclear topology. In particular, we apply our findings to the symmetric tensor algebra of a nuclear space. [ABSTRACT FROM AUTHOR]

We consider a version of the classical moment problem in the Beurling and Roumieu spaces of ultradifferentiable functions of mean type on the real axis. We obtain the necessary and sufficient conditions for the weights and under which, for each number sequence in the space generated by , there is an -ultradifferentiable function whose derivatives at zero coincide with the elements of the sequence. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we consider a problem of the similarity of complex symmetric operators to perturbations of restrictions of normal operators. For a subclass of cyclic complex symmetric operators in a finite-dimensional Hilbert space we prove the similarity to rank-one perturbations of restrictions of normal operators. The main tools are a truncated moment problem in C, and some objects similar to objects from the theory of spectral problems for Jacobi matrices. [ABSTRACT FROM AUTHOR]

With a proper function theoretic definition of the cloud of a positive measure with compact support in the real plane, a computational scheme of transforming the moments of the original measure into the moments of the uniformly distributed mass on the cloud is described. The main limiting operation involves exclusively truncated Christoffel-Darboux kernels, while error bounds depend on the spectral asymptotics of a Hankel kernel belonging to the Hilbert-Schmidt class. [ABSTRACT FROM AUTHOR]

This note is concerned with inverse wave scattering in one and two dimensional domains. It seeks to recover an unknown function based on measurements collected at the boundary of the domain. For one-dimensional problem, only one point of the domain is assumed to be accessible. For the two dimensional domain, the outer boundary is assumed to be accessible. It develops two iterative algorithms, in which an assumed initial guess for the unknown function is updated. The first method uses a set of sampling functions to formulate a moment problem for the correction to the assumed value. This method is applied to both one-dimensional and two dimensional domains. For two dimensional Helmholtz equation, it relies on a new effective filtering technique which is another contribution of the present work. The second method uses a direct formulation to recover the correction term. This method is only developed for the one-dimensional case. For all cases presented here, the correction to the assumed value is obtained by solving an over-determined linear system through the use of least-square minimization. Tikhonov regularization is also used to stabilize the least-square solution. A number of numerical examples are used to show their applicability and robustness to noise. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the full multidimensional moment problem for discrete measure using Vasilescu's idempotent approach based on Λ -multiplicative elements with respect to the associated square positive Riesz functional Λ. We give a sufficient condition for the existence of a discrete integral representation of the Riesz functional Λ , which turns out to be necessary in the bounded shift space case (in fact, it suffices to assume the density of polynomials in the corresponding L 2 -space). We pay special attention to Λ -multiplicative elements, providing several criteria guaranteeing that they are characteristic functions of single point sets. We also give an example showing that Λ -multiplicative elements may not be characteristic functions of single point sets. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with adjacency matrix A and spectral radius ρ. Let w k , ϕ k and ϕ k (i) be, respectively, the number walks of length k , closed walks of length k and closed walks starting and ending at vertex i after k steps. In this paper, we propose a measure-theoretic framework which allows us to relate walks in a graph with its spectral properties. In particular, we show that w k , ϕ k and ϕ k (i) can be interpreted as the moments of three different measures, all of them supported on the spectrum of A. Building on this interpretation, we leverage results from the classical moment problem to formulate a hierarchy of new lower and upper bounds on ρ , as well as provide alternative proofs to several well-known bounds in the literature. [ABSTRACT FROM AUTHOR]

Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindelöf locally compact Hausdorff space are given, and used to prove Stone–Weierstrass-type theorems for I . As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on I are uniquely determined by their restriction to various point-separating subsets of I . A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and without making use of spectral theory. [ABSTRACT FROM AUTHOR]

Abstract The Lindley distribution and its numerous generalizations are widely used in statistical and engineering practice. Recently, a power transformation of Lindley distribution, called the power Lindley distribution, has been introduced by M. E. Ghitany et al. who initiated the investigation of its properties and possible applications. In this article, new results on the power Lindley distribution are presented. The focus of this work is on the moment-(in)determinacy of the distribution for various values of the parameters. Afterwards, certain applications are provided to describe data sets of software metrics.