Indeterminate Moment Research Articles

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 −∞.

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.

A note on the general moment problem

In this note we show that given an indeterminate Hamburger moment sequence, it is possible to perturb the first moment in such way that the obtained sequence remains an indeterminate Hamburger moment sequence. As a consequence we prove that every sequence of real numbers is a moment sequence for a signed discrete measure supported in \(\mathbb{R}_+\).

Method of Mathematical Theory of Moments

Infinite matrices play an important role in many aspects of analysis, algebra, differential equations, andthetheory of mechanical vibrations.Jacobi matrices are interestingbecause they are the simplest representatives of symmetric operators in infinite-dimensional space. they are used in interpolation theory, quantum physics, moment problem.In this paper, basedon the elements of Jacobi matrix, itwill be determinedthe type of the operator that occurs when processing the results of measurements of random variables. The first type of operators arematrices,for which the moment problem has a unique solution, and Jacobi matrix generates a specific moment problem. The second type of operators arematrices, for which the moment problem has many solutions, and Jacobi matrix is said to generate an indeterminate moment problem.

Indeterminate moment problem associated with continuous dual q-Hahn polynomials

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.

Closable Hankel Operators and Moment Problems

In a paper from 2016 D. R. Yafaev considers Hankel operators associated with Hamburger moment sequences \(q_n\) and claims that the corresponding Hankel form is closable if and only if the moment sequence tends to 0. The claim is not correct, since we prove closability for any indeterminate moment sequence but also for certain determinate moment sequences corresponding to measures with finite index of determinacy. In an Erratum Yafaev (Integr Equ Oper Theory, to appear) has proved the result under quasi-analyticity assumptions, which imply that the moment sequence is determinate.

Estimates for the order of Nevanlinna matrices and a Berezanskii-type theorem

AbstractWe give an upper estimate for the order of the entire functions in the Nevanlinna parameterization of the solutions of an indeterminate Hamburger moment problem. Under a regularity condition this estimate becomes explicit and takes the form of a convergence exponent. Proofs are based on transformations of canonical systems and I.S.Kac' formula for the spectral asymptotics of a string. Combining with a lower estimate from previous work, we obtain a class of moment problems for which order can be computed. This generalizes a theorem of Yu.M.Berezanskii about spectral asymptotics of a Jacobi matrix (in the case that order is ⩽ 1/2).

Inverse of Infinite Hankel Moment Matrices

Let $(s_n)_{n\ge 0}$ denote an indeterminate Hamburger moment sequence and let $\mathcal H=\{s_{m+n}\}$ be the corresponding positive definite Hankel matrix. We consider the question if there exists an infinite symmetric matrix $\mathcal A=\{a_{j,k}\}$, which is an inverse of $\mathcal H$ in the sense that the matrix product $\mathcal A\mathcal H$ is defined by absolutely convergent series and $\mathcal A\mathcal H$ equals the identity matrix $\mathcal I$, a property called (aci). A candidate for $\mathcal A$ is the coefficient matrix of the reproducing kernel of the moment problem, considered as an entire function of two complex variables. We say that the moment problem has property (aci), if (aci) holds for this matrix $\mathcal A$. We show that this is true for many classical indeterminate moment problems but not for the symmetrized version of a cubic birth-and-death process studied by Valent and co-authors. We consider mainly symmetric indeterminate moment problems and give a number of sufficient conditions for (aci) to hold in terms of the recurrence coefficients for the orthonormal polynomials. A sufficient condition is a rapid increase of the recurrence coefficients in the sense that the quotient between consecutive terms is uniformly bounded by a constant strictly smaller than one. We also give a simple example, where (aci) holds, but an inverse matrix of $\mathcal H$ is highly non-unique.

Integral and series representations of q-polynomials and functions: Part I

By applying an integral representation for [Formula: see text], we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of [Formula: see text]-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include [Formula: see text]-Bessel functions, the Ramanujan function, Stieltjes–Wigert polynomials, [Formula: see text]-Hermite and [Formula: see text]-Hermite polynomials, and the [Formula: see text]-exponential functions [Formula: see text], [Formula: see text] and [Formula: see text]. Their representations are in turn used to derive many new identities involving [Formula: see text]-functions and polynomials. In this paper, we also present contour integral representations for the above mentioned functions and polynomials.

Dynamic ASEP, Duality, and Continuous q−1-Hermite Polynomials

Abstract We demonstrate a Markov duality between the dynamic asymmetric simple exclusion process (ASEP) and the standard ASEP. We then apply this to step initial data, as well as a half-stationary initial data (which we introduce). While investigating the duality for half-stationary initial data, we uncover and utilize connections to the continuous q−1-Hermite polynomials. Finally, we introduce a family of stationary initial data which are related to the indeterminate moment problem associated with these q−1-Hermite polynomials.

Symmetric moment problems and a conjecture of Valent

In 1998 Valent made conjectures about the order and type of certain indeterminate Stieltjes moment problems associated with birth and death processes which have polynomial birth and death rates of degree . Romanov recently proved that the order is as conjectured. We prove that the type with respect to the order is related to certain multi-zeta values and that this type belongs to the interval which also contains the conjectured value. This proves that the conjecture about type is asymptotically correct as . The main idea is to obtain estimates for order and type of symmetric indeterminate Hamburger moment problems when the orthonormal polynomials and those of the second kind satisfy and , where may be different, and and are positive constants. In this case the order of the moment problem is majorized by the harmonic mean of and . Here means that . This also leads to a new proof of Romanov's Theorem that the order is . Bibliography: 19 titles.

Bounds on Order of Indeterminate Moment Sequences

We investigate the order \(\rho \) of the four entire functions in the Nevanlinna matrix of an indeterminate Hamburger moment sequence. We give an upper estimate for \(\rho \) which is explicit in terms of the parameters of the canonical system associated with the moment sequence via its three-term recurrence. Under a weak regularity assumption, this estimate coincides with a lower estimate, and hence \(\rho \) becomes computable. Dropping the regularity assumption leads to examples where upper and lower bounds do not coincide and differ from the order. In particular, we provide examples for which the order is different from its lower estimate due to M.S. Livsic.

Holomorphic functions associated with indeterminate rational moment problems

We consider indeterminate rational moment problems on the real line with their associated orthogonal rational functions. There exists a Nevanlinna type parametrization relating to the problem, with associated Nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem. We prove that the growth at the essential singularities of the four functions in the Nevanlinna matrix is the same.

On the order of indeterminate moment problems

For an indeterminate moment problem we denote the orthonormal polynomials by Pn. We study the relation between the growth of the function P(z)=(∑n=0∞|Pn(z)|2)1/2 and summability properties of the sequence (Pn(z)). Under certain assumptions on the recurrence coefficients from the three term recurrence relation zPn(z)=bnPn+1(z)+anPn(z)+bn−1Pn−1(z), we show that the function P is of order α with 0<α<1, if and only if the sequence (Pn(z)) is absolutely summable to any power greater than 2α. Furthermore, the order α is equal to the exponent of convergence of the sequence (bn). Similar results are obtained for logarithmic order and for more general types of slow growth. To prove these results we introduce a concept of an order function and its dual.We also relate the order of P with the order of certain entire functions defined in terms of the moments or the leading coefficient of Pn.

Plancherel–Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems

We study the Plancherel–Rotach asymptotics of four families of orthogonal polynomials: the Chen–Ismail polynomials, the Berg–Letessier–Valent polynomials, and the Conrad–Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems, and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems.

A non-hyponormal operator generating Stieltjes moment sequences

A linear operator S in a complex Hilbert space H for which the set D∞(S) of its C∞-vectors is dense in H and {‖Snf‖2}n=0∞ is a Stieltjes moment sequence for every f∈D∞(S) is said to generate Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal operator S which generates Stieltjes moment sequences. What is more, D∞(S) is a core of any power Sn of S. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of indeterminate Stieltjes moment sequences. As a consequence, it is shown that there exists a non-hyponormal composition operator in an L2-space (over a σ-finite measure space) which is injective, paranormal and which generates Stieltjes moment sequences. The independence assertion of Barry Simonʼs theorem which parameterizes von Neumann extensions of a closed real symmetric operator with deficiency indices (1,1) is shown to be false.

A question by T. S. Chihara about shell polynomials and indeterminate moment problems

The generalized Stieltjes–Wigert polynomials depending on parameters 0 ≤ p < 1 and 0 < q < 1 are discussed. By removing the mass at zero of an N-extremal solution concentrated in the zeros of the D -function from the Nevanlinna parametrization, we obtain a discrete measure μ M , which is uniquely determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials ( P n M ) . As noticed by Chihara, these polynomials are the shell polynomials corresponding to the maximal parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well as the parameter sequence corresponding to the generalized Stieltjes–Wigert polynomials, and compute the value of related continued fractions. The mass points of μ M have been studied in recent papers of Hayman, Ismail–Zhang and Huber. In the special case of p = q , the maximal parameter sequence is constant and the determination of μ M and ( P n M ) gives an answer to a question posed by Chihara in 2001.

Density of polynomials in some [formula omitted] spaces on radial rays in the complex plane

In this paper, we study the necessary and sufficient conditions for the density of the linear space of matrix polynomials in a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays of the complex plane. The connection with a completely indeterminate Hamburger matrix moment problem is stated. Vector valued functions associated in a natural way with a function defined in the union of the radial rays are used. Thus, our first aim is the construction of a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays and a positive semi-definite matrix acting on the discrete part of the corresponding inner product. An isometric transformation which allows to reduce the problem of density to the case of the real line is introduced. Finally, some examples of such spaces are shown and its completeness is studied in detail.

The indeterminate moment problem for the [formula omitted]-Meixner polynomials

For a class of orthogonal polynomials related to the q -Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted L 2 -space explicitly. The result is proved in two ways; by a spectral decomposition of a suitable operator and by direct series manipulation. We discuss extensions to explicit non-positive measures and the relation to other indeterminate moment problems for the continuous dual q − 1 -Hahn and q -Laguerre polynomials.

Naimark extensions for indeterminacy in the moment problem. An example

Using the possibility of extending a symmetric operator to a selfadjoint one, which goes beyond the Hilbert space, we construct an example (in fact a class of them) of an indeterminate Hamburger moment sequence admitting a family of representing measures such that: the support of each of them is in arithmetic progression, the supports of all the measures together partition ℝ, none of them is N-extremal, and all of them are of infinite order. In fact, our goal is twofold: to present an example and to propose a method which may be useful for obtaining other results of this kind.