Strong Stieltjes Moment Problem Research Articles

Zeros of orthogonal Laurent polynomials and solutions of strong Stieltjes moment problems

The strong Stieltjes moment problem for a bisequence { c n } n = − ∞ ∞ consists of finding positive measures μ with support in [ 0 , ∞ ) such that ∫ 0 ∞ t n d μ ( t ) = c n for n = 0 , ± 1 , ± 2 , … . Orthogonal Laurent polynomials associated with the problem play a central role in the study of solutions. When the problem is indeterminate, the odd and even sequences of orthogonal Laurent polynomials suitably normalized converge in C ∖ { 0 } to distinct holomorphic functions. The zeros of each of these functions constitute (together with the origin) the support of two solutions μ ( ∞ ) and μ ( 0 ) . We discuss how odd and even subsequences of zeros of the orthogonal Laurent polynomials converge to the support points of μ ( ∞ ) and μ ( 0 ) .

Positivity of Continued Fractions Associated with Rational Stieltjes Moment Problems

The constructive solution of the strong Stieltjes moment problem can be linked with positive Perron-Caratheodory continued fractions (PC-fractions) which are contradictions of positive Thron continued fractions (T-fractions). Their approximants are two-point Pade approximants for a related function. The multi-point moment problem similarly leads to positive multi-point Pade continued fractions (MP-fraction) and positive extended multi-point Pade continued fractions (EMP-fraction) whose approximants are multi-point Pade approximants. These relationships are explored also in the situation where the positivity condition is dropped.

Para-orthogonal Laurent polynomials and the strong Stieltjes moment problem

On the space, A of Laurent polynomials we consider a linear functional L which is positive definite on (0,∞) and is defined in terms of a given bisequence, { c k } k=−∞ ∞. For each ω>0, we define a sequence { N n ( z, ω)} n=0 ∞ of rational functions in terms of two sequences of orthogonal Laurent polynomials, { Q n ( z)} n=0 ∞ and { Q ̂ n(z)} n=0 ∞ , which span A in the order {1, z −1, z, z −2, z 2,…} and {1, z, z −1, z 2, z −2,…}, respectively. It is shown that the numerators and denominators of each N n ( z, ω) are linear combinations of the canonical numerators and denominators of a modified PC-fraction. Consequently, { N 2 n ( z, ω)} n=0 ∞ and { N 2 n+1 ( z, ω)} n=0 ∞ converge uniformly on compact subsets of C –{0} to analytic functions and hence lead to additional solutions to the strong Stieltjes moment problem.

Nevanlinna matrices for the strong Stieltjes moment problem

Let ∗c n∗ n=−∞ ∞ be a doubly infinite sequence of real numbers. A solution of the strong Hamburger moment problem is a positive measure σ on (−∞, ∞) such that c n= ∫ −∞ ∞ u n dδ(u) for n = 0, ±1, ±2,…. A solution of the strong Stieltjes moment problem is a positive measure σ on [0, ∞) such that c n= ∫ 0 ∞ u n dδ(u) for n = 0, + 1 ̄ , + 2 ̄ ,… . A moment problem is indeterminate if there exists more than one solution. With an indeterminate strong Hamburger moment problem there is associated a Nevanlinna matrix of functions α, β, γ, δ holomorphic in ofC − ∗0∗. These functions have growth properties partly similar to properties of analogous entire functions associated with an indeterminate classical Hamburger moment problem. In this paper we obtain a stronger growth result in the case where the strong Stieltjes moment problem is solvable.

Power moments of negative order for the principal spectral function of a string

We established necessary and sufficient conditions for the existence of finite power moments of all integer negative orders for the principal spectral function of a string. The necesity of this problem is explained by its relation to the so-called strong Stieltjes moment problem.

Extremal solutions of the strong Stieltjes moment problem

A solution of the strong Stieltjes moment problem for the sequence { C n : n = o,±1, ±2,…} is a finite positive measure μ on [0, ∞) such that c n= ∫ 0 ∞ t ndμ(t) for all n, while a solution of the strong Hamburger moment problem for the same sequence is a finite positive measure μ on (−∞, ∞) such that c n= ∫ −∞ ∞ t ndμ(t) for all n. When the Hamburger problem is indeterminate, there exists a one-to-one correspondence between all solutions μ and all Nevanlinna functions ϕ, the constant ∞ included. The correspondence is given by Fμ(z)=− ∝(z)ϕ(z)−γ(z) β(z)ϕ(z)−δ(z) , where α, β, γ, δ are certain functions holomorphic in C − {0} . The extremal solutions are the solutions μ t corresponding to the constant functions ϕ( z) ≡ t, t ϵ R ∪ {∞} . The accumulation points of the (isolated) set Z t of zeros of β( z) t − δ( z) consists of 0 and ∞. The support of the extremal solution μ t is the set Z t ∪ {0}. There exists an interval [ t (0), t (∞)] such that the extremal solutions of the Stieltjes problem are exactly those μ t for which t ∈ [ t (0), t (∞)]. The measures μ t (0) and μ t (∞) are natural solutions, and the only ones. If ξ k ( n) denote the zeros of the orthogonal Laurent polynomials determined by { C n } ordered by size, then { ξ k ( n) } tends to 0 and { ξ n− k ( n) } tends to ∞ for arbitrary constant k when n tends to ∞.

Para-orthogonal Laurent polynomials and associated sequences of rational functions

On the space, Л, of Laurent polynomials (L-polynomials) we consider a linear functional ℒ which is positive definite on (0, ∞) and is defined in terms of a given bisequence, {μ k } −∞ ∞ . Two sequences of orthogonal L-polynomials, {Q n (z) 0 ∞ and $$\{ \hat Q_n (z)\} _0^\infty $$ , are constructed which span Λ in the order {1,z −1,z,z −2,z 2,...} and {1,z,z −1,z 2,z −2,...} respectively. Associated sequences of L-polynomials {P n (z) 0 ∞ , and $$\{ \hat P_n (z)\} _0^\infty $$ are introduced and we define rational functions $$M_n (z,w) = (P_n (z) + w\hat P_n (z))/(Q_n (z) + w\hat Q_n (z))$$ , wherew is a fixed positive number. The partial fraction decomposition and integral representation of,M n (z, w) are given and correspondence of {M n (z, w)} is discussed. We get additional solutions to the strong Stieltjes moment problem from subsequences of {M n (z, w)}. In particular when {μ k } −∞ ∞ is a log-normal bisequence, {M 2n (z, w)} and {M 2n+1 (z, w)} yield such solutions.

Separate convergence of general 𝑇-fractions

This article is concerned with the separate convergence of the sequences of numerators { A n ( z ) } \{ {A_n}(z)\} and denominators { B n ( z ) } \{ {B_n}(z)\} of the approximants A n ( z ) / ( B n ( z ) {A_n}(z)/({B_n}(z) of the general T {\text {T}} -fraction \[ K\limits _{n = 1}^\infty \left ( {\frac {{{F_n}z}}{{1 + {G_n}z}}} \right ).\] Convergence results for sequences { A n ( z ) / Γ n ( z ) } \{ {A_n}(z)/{\Gamma _n}(z)\} and { B n ( z ) / Γ n ( z ) } \{ {B_n}(z)/{\Gamma _n}(z)\} , where the sequence { Γ n ( z ) } \{ {\Gamma _n}(z)\} is "sufficiently simple" are also derived.

A Strong Stieltjes Moment Problem

This paper is concerned with double sequences of complex numbers $C = \{ {c_n}\} _{ - \infty }^\infty$ and with formal Laurent series ${L_0}(C) = \Sigma _1^\infty - {c_{ - m}}{z^m}$ and ${L_\infty }(C) = \Sigma _0^\infty {c_m}{z^{ - m}}$ generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having ${L_0}(C)$ and ${L_\infty }(C)$ as asymptotic expansions at $z = 0$ and $z = \infty$, respectively? (2) Does there exist a real-valued bounded, monotonically increasing function $\psi (t)$ with infinitely many points of increase on $[0, \infty )$ such that, for every integer n, ${c_n} = \int _0^\infty {{{( - t)}^n} d\psi (t)}$? The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function $\psi (t)$ has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive T-fractions) which correspond to ${L_0}(C)$ at $z = 0$ and ${L_\infty }(C)$ at $z = \infty$. Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have ${L_0}(C)$ and ${L_\infty }(C)$ as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions ${\psi ^{(0)}}(t)$ and ${\psi ^{(1)}}(t)$ solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive T-fraction that corresponds to both ${L_0}(C)$ and ${L_\infty }(C)$. References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.

A strong Stieltjes moment problem

This paper is concerned with double sequences of complex numbers C = { c n } − ∞ ∞ C\, = \,\{ {c_n}\} _{ - \infty }^\infty and with formal Laurent series L 0 ( C ) = Σ 1 ∞ − c − m z m {L_0}(C)\, = \,\Sigma _1^\infty \, - \,{c_{ - m}}{z^m} and L ∞ ( C ) = Σ 0 ∞ c m z − m {L_\infty }(C)\, = \,\Sigma _0^\infty \,{c_m}{z^{ - m}} generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having L 0 ( C ) {L_0}(C) and L ∞ ( C ) {L_\infty }(C) as asymptotic expansions at z = 0 z\, = \,0 and z = ∞ z\, = \,\infty , respectively? (2) Does there exist a real-valued bounded, monotonically increasing function ψ ( t ) \psi (t) with infinitely many points of increase on [ 0 , ∞ ) [0,\,\infty ) such that, for every integer n, c n = ∫ 0 ∞ ( − t ) n d ψ ( t ) {c_n}\, = \,\int _0^\infty {{{( - t)}^n}\,d\psi (t)} ? The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function ψ ( t ) \psi (t) has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive T-fractions) which correspond to L 0 ( C ) {L_0}(C) at z = 0 z\, = \,0 and L ∞ ( C ) {L_\infty }(C) at z = ∞ z\, = \,\infty . Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have L 0 ( C ) {L_0}(C) and L ∞ ( C ) {L_\infty }(C) as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions ψ ( 0 ) ( t ) {\psi ^{(0)}}(t) and ψ ( 1 ) ( t ) {\psi ^{(1)}}(t) solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive T-fraction that corresponds to both L 0 ( C ) {L_0}(C) and L ∞ ( C ) {L_\infty }(C) . References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.