The moment problem associated with the Stieltjes-Wigert polynomials

We consider the indeterminate Stieltjes moment problem associated with the Stieltjes-Wigert polynomials. After a presentation of the well-known solutions, we study a transformation T of the set of solutions. All the classical solutions turn out to be fixed under this transformation but this is not the case for the so-called canonical solutions. Based on generating functions for the Stieltjes-Wigert polynomials, expressions for the entire functions A, B, C, and D from the Nevanlinna parametrization are obtained. We describe T⁽ⁿ⁾(μ) for when μ=μ₀ is a particular N-extremal solution and explain in detail what happens when n→∞.     

The tracial moment problem and trace-optimization of polynomials

The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace ${f(\underline {A})}$ can attain for a tuple of matrices ${\underline {A}}$ ? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives effectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satisfies a certain condition called flatness, then the relaxation is exact. In this case it is shown how to extract global trace-optimizers with a procedure based on two ingredients. The first is the solution to the truncated tracial moment problem, and the other crucial component is the numerical implementation of the Artin-Wedderburn theorem for matrix *-algebras due to Murota, Kanno, Kojima, Kojima, and Maehara. Trace-optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side—two topics with many applications, the most prominent being to linear systems engineering and quantum physics. The optimization problems discussed here facilitate new possibilities for applications, e.g. in operator algebras and statistical physics.     

Trace-positive polynomials and the quartic tracial moment problem

The tracial analog of Hilbert's classical result on positive binary quartics is presented: a trace-positive bivariate noncommutative polynomial of degree at most four is a sum of hermitian squares and commutators. This is applied via duality to investigate the truncated tracial moment problem: a sequence of real numbers indexed by words of degree four in two noncommuting variables with values invariant under cyclic permutations of the indexes, can be represented with tracial moments of matrices if the corresponding moment matrix is positive definite. Understanding trace-positive polynomials and the tracial moment problem is one of the approaches to Connes' embedding conjecture.     

Orthogonal Laurent polynomials and the strong Hamburger moment problem

This paper is concerned with the strong Hamburger moment problem (SHMP): For a given double sequence of real numbers C = {c_n}^∞_?∞, does there exist a real-valued, bounded, non-decreasing function ψ on (?∞, ∞) with infinitely many points of increase such that for every integer n, c_n = ∝^∞_?∞ (?t)ⁿdψ(t)? Necessary and sufficient conditions for the existence of such a function ψ are given in terms of the positivity of certain Hankel determinants associated with C. Our approach is made through the study of orthogonal (and quasi-orthogonal) Laurent polynomials (referred to here as L-polynomials) and closely related Gaussian-type quadrature formulas. In the proof of sufficiency an inner product for L-polynomials is defined in terms of the given double sequence C. Since orthogonal L-polynomials are believed to be of interest in themselves, some examples of specific systems are considered.     

Szegő polynomials and the truncated trigonometric moment problem

We show how Szegő polynomials can be used in the theory of truncated trigonometric moment problem. Mathematics Subject Classification Primary—42A70; Secondary—42C15 The work was done during a visit of the first author to UNESP with a fellowship from FAPESP in September–October, 2002. The research of the second author was supported by grants from CNPq and FAPESP of Brazil.     

Co-Recursivity and karlin-McGregor duality for indeterminate moment problems

We analyze co-recursivity for indeterminate Hamburger moment problems and the duality transformation of Karlin and McGregor for indeterminate Stieltjes moment problems. In both cases the transformed Nevanlinna matrix is given and the Nevanlinna extremal measures are discussed. An example involving associated polynomials, relevant for a quartic birth and death process, is worked out.     

Hausdorff's moment problem and expansions in Legendre polynomials

A new proof is given for Hausdorff's condition on a set of moments which determines when the function generating these moments is in L². The proof uses Legendre polynomials and their discrete extensions found by Tchebychef. Then an extension is given to a weighted L² space using Jacobi polynomials and their discrete extensions.     

The Sobolev-type moment problem

We propose necessary and sufficient conditions for a bisequence of complex numbers to be a moment one of Sobolev type over the real line, the unit circle and the complex plane. We achieve this through converting the moment problem in question into a matrix one and utilizing some techniques coming from operator theory. This allows us to consider the Sobolev type moment problem in its full generality, not necessarily in the diagonal case and even of infinite order.     

The Bezoutian and the eigenvalue-separation problem for matrix polynomials

A generalized Bezout matrix for a pair of matrix polynomials is studied and, in particular, the structure of its kernel is described and the relations to the greatest common divisor of the given matrix polynomials are presented. The classical root-separation problems of Hermite, Routh-Hurwitz and Schur-Cohn are solved for matrix polynomials in terms of this Bezout matrix. The eigenvalue-separation results are also expressed in terms of Hankel matrices whose entries are Markov parameters of rational matrix function. Some applications of Jacobi's method to these problems are pointed out.     

The moment problem on the Wiener space

Consider an L¹-continuous functional ? on the vector space of polynomials of Brownian motion at given times, suppose ? commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, , mapping the Wiener space to R.In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which ? can be written in the form ∫⋅dμ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables are nonnegative.     

An extremal problem for polynomials

We give a solution to an extremal problem for polynomials, which asks for complex numbers α₀,…,α_n${\alpha }_0,\dots ,{\alpha }_n$ of unit magnitude that minimise the largest supremum norm on the unit circle for all polynomials of degree n whose k  -th coefficient is either α_k${\alpha }_k$ or −α_k$-{\alpha }_k$.     

Smoothed projection methods for the moment problem

Summary We discuss the problem of reconstructing a functionf from a finite set of moments. Problems of this kind typically arise as discretizations of integral equations of the first kind. We propose an algorithm which is based on a pointwise optimization of the pointspread function, which makes it particularly suitable for local reconstructions. The method is compared with known methods as Backus-Gilbert and projection methods. Convergence of the method is proved and the rate of convergence is determined. The influence of noisy data is examined and numerical examples show the usefulness of the method.