This paper presents a bioinspired optimization approach to address a class of inverse problems involving entropy optimization (EOP) from knowledge of the moments of a distribution function. In particular, we study the Hausdorff moment problem, where one seeks to reconstruct a (probability) density distribution by inverting a completely monotonic sequence of moments of the distribution in a bounded interval. It is shown that the resulting EOP can be handled very efficiently using the collective intelligence of a swarm (of optimizers), which provides a robust and accurate solution by effectively incorporating information from up to a thousand moments of the density. The efficacy of the approach is demonstrated by reconstructing the invariant density functions for the logistics map, spectral densities of large real-symmetric random matrices, encountered in the study of physics of disordered solids, and financial time series involving daily price fluctuations of a mutual fund. The agreement between true densities and the corresponding maximum-entropy approximants is examined by comparing the Kullback-Leibler divergence and the Fisher information of the densities.

Abstract The discrete data encoded in the power moments of a positive measure, fast decaying at infinity on Euclidean space, are incomplete for recovery, leading to the concept of moment indeterminateness. On the other hand, classical integral transforms (Fourier‐Laplace, Fantappiè, Poisson) of such measures are complete, often invertible via an effective inverse operation. The gap between the two non‐uniqueness/uniqueness phenomena is manifest in the dual picture, when trying to extend the measure, regarded as a positive linear functional, from the polynomial algebra to the full space of continuous functions. This point of view was advocated by Marcel Riesz a century ago, in the single real variable setting. Notable advances in functional analysis have their root in Riesz's celebrated four notes devoted to the moment problem. A key technical ingredient being there the monotone approximation by polynomials of kernels of integral transforms. With inherent new obstacles, we reappraise in the context of several real variables M. Riesz's variational principle. The result is an array of necessary and sufficient moment indeterminateness criteria, some raising real algebra questions, as well as others involving intriguing analytic problems, all gravitating around the concept of moment separating function.

Time Optimality and the Markov Moment Problem: A General Framework for Linear and Nonlinear Cases

The first aim of this study is to point out new aspects of approximation theory applied to a few classes of holomorphic functions via Vitali’s theorem. The approximation is made with the aid of the complex moments of the functions involved, which are defined similarly to the moments of a real-valued continuous function. By applying uniform approximation of continuous functions on compact intervals via Korovkin’s theorem, the hard part concerning uniform approximation on compact subsets of the complex plane follows according to Vitali’s theorem. The theorem on the set of zeros of a holomorphic function is also applied. In the end, the existence and uniqueness of the solution for a multidimensional moment problem are characterized in terms of limits of sums of quadratic expressions. This is the application appearing at the end of the title. Consequences resulting from the first part of the paper are pointed out with the aid of functional calculus for self-adjoint operators.

In the inverse Stieltjes moment problem, one seeks to reconstruct a non-negative distribution from its spectral moments defined on an unbounded interval 〈0,∞. In chemical physics, this problem arises when computing continuous quantities such as photoionization cross sections or electronic decay widths using discretized approximations to the electronic continuum. While Stieltjes imaging (SI) is the established method in this context, it provides only sparse, discrete sampling of the distribution. Here, we develop a maximum entropy (ME) approach to the solution of the inverse Stieltjes moment problem in the context of Fano theory of resonances, where the sought-after quantity is the decay width function. We implement two ME variants-with polynomial and exponential asymptotic damping-and introduce an averaging procedure over spectral moment orders that addresses convergence issues and provides reliable error estimates. Benchmarking against abinitio Fano-ADC data for molecular Auger decay and interatomic Coulombic decay, we show that ME achieves comparable accuracy to SI while providing a continuous representation of the function. Our results establish ME as a valuable alternative to SI, particularly when analytical continuation or additional verification is required.

We derive a Binet-type formula for operator-valued sequences satisfying linear recurrence relations, extending the classical scalar case to the setting of bounded operators on Hilbert spaces. In this framework, we analyze the operator moment problem as an application, establishing new connections between recursive operator sequences and moment sequences.

The scalar moment problem was first introduced by T. J. Stieltjes in his work ``Recherches sur les fractions continues'' Annals of the Faculty of Sciences of Toulouse 8, 1--122, (1895). He formulated it as follows: Given the moments of order $k$ ($k=0,1,2,\dots$), find a positive mass distribution on the half-line $[0,+\infty)$. The study of matrix and operator moment problems was initiated by M. G. Krein in his seminal paper ``Fundamental aspects of the representation theory of Hermitian operators with deficiency index $(m,m)$'' Translations of the American Mathematical Society, Series II, 97, 75--143, (1949). This paper is related to the truncated Hausdorff matrix moment (THMM) problem: the truncated moment problem on a compact interval $[a,b]$ in contrast to the Stieltjes moment problem on $[0,+\infty)$ and the Hamburger moment problem on $(-\infty,+\infty)$. Our approach relies on V. P. Potapov’s method, which reformulates interpolation and moment problems as equivalent matrix inequalities and introduces auxiliary matrices that satisfy the $\widetilde{J}_q$--inner function property of the Potapov class, together with a system of column pairs. The method begins by constructing Hankel matrices from the prescribed moments. If these matrices are positive semidefinite, the THMM problem is solvable. In the strictly positive definite case, known as the non-degenerate case, we transform the associated matrix inequalities to derive the Nevanlinna (or resolvent) matrix of the THMM problem, which characterizes its solutions. This framework has been extensively applied, for instance in A. E. Choque Rivero, Yu. M. Dyukarev, B. Fritzsche, and B. Kirstein, ``A truncated matricial moment problem on a finite interval'', in Interpolation, Schur Functions and Moment Problems, Operator Theory: Advances and Applications, Birkh\"{a}user, Basel, 165, 121--173, (2006). The main contribution of the present work is to represent the Nevanlinna matrix of the THMM problem in terms of orthogonal matrix polynomials (OMP) and their associated polynomials of the second kind at point $b$. Note that the representation at point $a$ was obtained earlier in A. E. Choque Rivero, ``From the Potapov to the Krein–Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem'' Bulletin of the Mexican Mathematical Society, 21(2), 233--259 (2015). In addition, we establish new identities involving OMP and reformulate an explicit relationship between the Nevanlinna matrices of the THMM problem at points $a$ and $b$, through OMP.

We study sequences of bounded operators \((T_n)_{n \ge 0}\) on a complex separable Hilbert space \(\mathcal{H}\) that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients \(A_0, A_1, \dots, A_{r-1}\) are pairwise commuting bounded operators on \(\mathcal{H}\). \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for \(T_n\). As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

A new pair of transformations and applications to generalized informational inequalities and Hausdorff moment problem

We study an exact controllability problem for a system governed by the heat equation with Neumann boundary control and boundary noise. We reduce the control problem to a moment problem for which we establish sufficient conditions for its resolution.

Abstract The truncated multidimensional moment problem is studied as an interpolation problem on the special sets, which will be defined in this paper. The step-by-step algorithm is obtained and description of all solutions in symmetric form is found.

In connection with the non-uniqueness of the Stieltjes moment problem on $(0,\infty)$, Stieltjes constructed the nontrivial function $ f(x)=e^{-x^{1/4}}\sin\!\big(x^{1/4}\big) $ satisfying $\int_{0}^{\infty} x^{n}f(x)\,dx=0$ for all integers $n\ge 0$. We extend this by considering \[ I(k)=\int_{0}^{\infty} e^{-x^{1/4}}\sin\!\big(x^{1/4}\big)\,x^{k}\,dx \] for real $k\ge 0$, evaluating $I(k)$ explicitly and proving $I(k)=0$ if and only if $k\in\mathbb{Z}_{\ge 0}$. More generally, for parameters $m>0$, $\alpha>0$, $\beta\in\mathbb{R}$, $q,k\in\mathbb{R}$ we analyze \[ I_{m,q}(k;\alpha,\beta)=\int_{0}^{\infty} e^{-\alpha x^{1/m}}\sin\!\big(\beta x^{1/m}\big)\,x^{k}(x^{1/m})^{q}\,dx, \] derive a closed form, and give necessary and sufficient conditions for its vanishing. We also establishcosine analogues, both for the Stieltjes example and for the generalized integral mentioned above.As a consequence, we obtain integral representations of Γ(A) for suitable A > 0, as well as integralformulas for several classical constants arising from gamma function. To understand the importanceof integrals that vanish for every value of a continuous parameter, we will also discuss Salem’sequivalence of the Riemann hypothesis, which is formulated in terms of such a parameter-dependentintegral.

Exact solving approach to moment problems with nonnegative Chebyshev ambiguity sets

We provide new conditions under which the alternating projection sequence converges in norm for the convex feasibility problem where a linear subspace with finite codimension N ≥ 2 N\geq 2 and a lattice cone in a Hilbert space are considered. The first result holds for any Hilbert lattice, assuming that the orthogonal complement of the linear subspace admits a basis made by disjoint vectors with respect to the lattice structure. The second result is specific for ℓ 2 ( N ) \ell ^2(\mathbb {N}) and is proved when only one vector of the basis is not in the cone but the sign of its components is definitively constant and its support has finite intersection with the supports of the remaining vectors.

A bstract We study the operator product expansion (OPE) of identical scalars in a conformal four-point correlator as a Stieltjes moment problem, and use Riemann-Liouville type fractional differential operators to generate classical moments from the correlation function. We use crossing symmetry to derive leading and subleading relations between moments in ∆ and J 2 ≡ ℓ ( ℓ + d − 2) in the “heavy” limit of large external scaling dimension, and combine them with constraints from unitarity to derive two-sided bounds on moment sequences in ∆ and the covariance between ∆ and J 2 . The moment sequences which saturate these bounds produce “saddle point” solutions to the crossing equations which we identify as particular limits of correlators in a generalized free field (GFF) theory. This motivates us to study perturbations of heavy GFF four-point correlators by way of saddle point analysis, and we show that saddles in the OPE arise from contributions of fixed-length operator families encoded by a decomposition into higher-spin conformal blocks. To apply our techniques, we consider holographic correlators of four identical single scalar fields perturbed by a bulk interaction, and use their first few moments to derive Gaussian weight-interpolating functions that predict the OPE coefficients of interacting double-twist operators in the heavy limit.

Necessary and sufficient conditions for the existence of the solutions of a class of scalar and mainly for operator-valued moment problems are reviewed. This was the first motivation for proving our constrained extension results for linear operators. Polynomial approximations on bounded and on unbounded closed subsets are very useful in proving the uniqueness of the solution. We also reviewed earlier results on the extension of positive linear functional and operators. Such results are applied to ensure the extension of our linear solution from the subspace of polynomials to a larger function space. In most of the cases from below, this is made using polynomial approximation in one and several variables. Besides positivity, our solution is bounded from above by a dominating linear, sublinear or only convex continuous operator, on the entire domain space or only on its positive cone. This allows estimating the norm of the linear solution.

On the moment problem and the linear difference equations with periodic coefficients

Motivated by applications in quantum information, we will consider the problem of establishing necessary and sufficient conditions so that linear functionals L : R n [ x 1 , … , x d ] → R L: \mathbb {R}_n[x_1, \ldots , x_d] \to \mathbb {R} with L ( 1 ) = 1 L(1) = 1 , where d ≤ n d \leq n , have a representing measure on vertices of the hypercube in R d \mathbb {R}^d , i.e., there exists a probability measure μ \mu such that L ( p ) = ∫ R d p ( x ) d μ ( x ) ≔ ∫ ⋯ ∫ R d p ( x 1 , … , x d ) d μ ( x 1 , … , x d ) \begin{equation*} L(p) = \int _{\mathbb {R}^d} p(x) \, d\mu (x) ≔\int \cdots \int _{\mathbb {R}^d} p(x_1, \ldots , x_d) d\mu (x_1, \ldots , x_d) \end{equation*} for p ∈ R n [ x 1 , … , x d ] p \in \mathbb {R}_n[x_1, \ldots , x_d] and s u p p μ ⊆ K d supp\, \mu \subseteq K_d where K d ≔ { ( k 1 , … , k d ) ∈ R d : k j = ± 1 , for j = 1 , … , d } K_d≔\{(k_1,\ldots ,k_d) \in \mathbb {R}^d : \ k_j=\pm 1, \text { for } j=1,\dots ,d\} . Taking advantage of the group structure of K d K_d and some tools from algebraic geometry and functional analysis, we will show that concrete necessary and sufficient conditions can be communicated in terms of certain consistency conditions and also a system of inequalities which are known in the quantum information theory literature as Bell inequalities. A variant of this problem (which is highly relevant to corresponding Bell nonlocality problems in quantum information theory), when L : A → R L: \mathcal {A} \to \mathbb {R} , where A ⊆ R n [ x 1 , … , x d ] \mathcal {A} \subseteq \mathbb {R}_n[x_1, \ldots , x_d] is a subspace, will also be considered. Finally, we will provide a formulation of the aforementioned concrete solution in a finite multiplicative Abelian group setting, which may be of independent interest.

Toeplitz, Hankel, de Branges and two truncated matrix moment problems

The main result of the paper is an interesting relation between the solution of the truncated exponential moment problem and truncated classical moment problem, considered on the half-line or on a compact interval.