Dynamic inverse problem for special system associated with Jacobi matrices and classical moment problems

TL;DR

This paper presents a unified dynamical systems approach to classical moment problems, linking inverse spectral data with Jacobi matrices and providing new insights into existence and uniqueness of solutions.

Contribution

It introduces a novel unified method based on dynamical systems and boundary control to solve and analyze classical moment problems using Jacobi matrices.

Findings

Solutions are characterized by spectral measures of Jacobi matrices.

Existence conditions are expressed via Hankel matrices.

Uniqueness is established through Krein-type equations.

Abstract

We consider the Hamburger, Stieltjes and Hausdorff moment problems, that are problems of the construction of a Borel measure supported on a real line, on a half-line or on an interval $(0,1)$, from a prescribed set of moments. We propose a unified approach to these three problems based on using the auxiliary dynamical system with the discrete time associated with a semi-infinite Jacobi matrix. It is show that the set of moments determines the inverse dynamic data for such a system. Using the ideas of the Boundary Control method for every $N\in \mathbb{N}$ we can recover the spectral measure of a $N\times N$ block of Jacobi matrix, which is a solution to a truncated moment problem. This problem is reduced to the finite-dimensional generalized spectral problem, whose matrices are constructed from moments and are connected with well-known Hankel matrices by simple formulas. Thus the results on existence of solutions to Hamburger, Stieltjes and Hausdorff moment problems are naturally given in terms of these matrices. We also obtain results on uniqueness of the solution of moment problems, where as a main tool we use the Krein-type equations of inverse problem.

Equations

$$s_k={\int }_{-\infty }^{\infty }{\lambda }^{k}d\rho (\lambda ),k=0,1,2,\dots .$$

$$A=\left(\begin{array}{cccccc}{b}_1& a_1& 0& 0& 0& \dots \\ a_1& b_2& a_2& 0& 0& \dots \\ 0& a_2& b_3& a_3& 0& \dots \\ \dots & \dots & \dots & \dots & \dots & \dots \end{array}\right).$$

$$\left\{\begin{array}[]{l}v_{n,t+1}+v_{n,t-1}-a_{n}v_{n+1,t}-a_{n-1}v_{n-1,t}-b_{n}v_{n,t}=0,\,\,t\in\mathbb{N}_{0},\,\,n\in 1,\ldots,N,\\ v_{n,\,-1}=v_{n,\,0}=0,\quad n=1,2,\ldots,N+1,\\ v_{0,\,t}=f_{t},\quad v_{N+1,\,t}=0,\quad t\in\mathbb{N}_{0},\end{array}\right.$$

$$(A^N\psi {)}_n=\{\begin{array}{l}{b}_1{\psi }_1+a_1{\psi }_2,n=1,\\ a_n{\psi }_{n+1}+a_{n-1}{\psi }_{n-1}+b_n{\psi }_n,2⩽n⩽N-1,\\ a_{N-1}{\psi }_{N-1}+b_N{\psi }_N,n=N,\end{array}$$

$$\{\begin{array}{l}{a}_n{\varphi }_{n+1}+a_{n-1}{\varphi }_{n-1}+b_n{\varphi }_n=\lambda {\varphi }_n,n⩾1,\\ {\varphi }_0=0,{\varphi }_1=1.\end{array}$$

$$({\varphi }^k,{\varphi }^l)={\delta }_{kl}{\rho }_k,k,l=1,\dots ,N,$$

$$\{{\lambda }_k,{\rho }_k{\}}_{k=1}^N$$

$$c_t=\sum _{s=0}^t{f}_s{g}_{t-s},t\in \mathbb{N}\cup \{0\}.$$

$$\left\{\begin{array}[]{l}\mathcal{T}_{t+1}+\mathcal{T}_{t-1}-\lambda\mathcal{T}_{t}=0,\\ \mathcal{T}_{0}=0,\,\,\mathcal{T}_{1}=1.\end{array}\right.$$

$$v_{n,t}^f=\{\begin{array}{l}{\sum }_{k=1}^N{c}_t^k{\varphi }_n^k,n=1,\dots ,N,\\ f_t,n=0.\end{array},c^k=\frac{1}{{\rho }_k}\mathcal{T}\left({\lambda }_k\right)\ast f.$$

$$W_N^{T}f:=v_{n,T}^f,n=1,\dots ,N.$$

$${\left(R_N^{T}f\right)}_t=v_{1,t}^f,t=1,\dots ,T.$$

$$\left(R_N^{T}f\right)=r^N\ast f_{\cdot -1},$$

$${\left(C_N^{T}f,g\right)}_{{\mathcal{F}}^T}={\left(v_{\cdot ,T}^f,v_{\cdot ,T}^g\right)}_{{\mathcal{H}}^N}={\left(W_N^{T}f,W_N^{T}g\right)}_{{\mathcal{H}}^N},C_N^T={\left(W_N^T\right)}^{\ast }{W}_N^T.$$

$$r_{t-1}^N={\left(R_N^T\delta \right)}_t=v_{1,t}^{\delta },t=1,\dots .$$

$${\rho }^N(\lambda )=\sum _{\{k|{\lambda }_k<\lambda \}}\frac{1}{{\rho }_k},$$

$$v_{n,t}^f={\int }_{-\infty }^{\infty }\sum _{k=1}^t{\mathcal{T}}_k(\lambda )f_{t-k}{\phi }_n(\lambda )d{\rho }^N(\lambda ),n,t\in \mathbb{N},$$

$$r_{t-1}^N={\int }_{-\infty }^{\infty }{\mathcal{T}}_t(\lambda )d{\rho }^N(\lambda ),t\in \mathbb{N},$$

$$\{C_N^T{\}}_{l+1,m+1}={\int }_{-\infty }^{\infty }{\mathcal{T}}_{T-l}(\lambda ){\mathcal{T}}_{T-m}(\lambda )d{\rho }^N(\lambda ),l,m=0,\dots ,T-1.$$

$$\left\{\begin{array}[]{l}u_{n,\,t+1}+u_{n,\,t-1}-a_{n}u_{n+1,\,t}-a_{n-1}u_{n-1,\,t}-b_{n}u_{n,\,t}=0,\quad n\in\mathbb{N},\,t\in\mathbb{N}_{0},\\ u_{n,\,-1}=u_{n,\,0}=0,\quad n\in\mathbb{N},\\ u_{0,\,t}=f_{t},\quad t\in\mathbb{N}_{0},\end{array}\right.$$

$$u_{n,t}^f=\prod _{k=0}^{n-1}{a}_k{f}_{t-n}+\sum _{s=n}^{t-1}{w}_{n,s}{f}_{t-s-1},n,t\in \mathbb{N},$$

$${\left(R^{T}f\right)}_t=u_{1,t}^f,t=1,\dots ,T.$$

$${\left(R^{T}f\right)}_t=u_{1,t}^f=f_{t-1}+\sum _{s=1}^{t-1}{w}_{1,s}{f}_{t-1-s}t=1,\dots ,T.$$

$$\left(R^{T}f\right)=r\ast f_{\cdot -1}.$$

$${\left(R^T\delta \right)}_t=u_{1,t}^{\delta }=r_{t-1}.$$

$${\left(W^{T}f\right)}_n:=u_{n,T}^f,n=1,\dots ,T.$$

$${\left(W^{T}f\right)}_n=u_{n,T}^f=\prod _{k=0}^{n-1}{a}_k{f}_{T-n}+\sum _{s=n}^{T-1}{w}_{n,s}{f}_{T-s-1},n=1,\dots ,T.$$

$$W^{T}f=\left(\begin{array}{c}{u}_{1,T}\\ u_{2,T}\\ \cdot \\ u_{k,T}\\ \cdot \\ u_{T,T}\end{array}\right)=\left(\begin{array}{cccccc}{w}_{1,T-1}& w_{1,T-2}& w_{1,T-3}& \dots & \dots & 1\\ w_{2,T-1}& w_{2,T-1}& \dots & \dots & a_1& 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ w_{k,T-1}& \dots & {\prod }_{j=0}^{k-1}{a}_j& 0& \dots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ {\prod }_{k=0}^{T-1}{a}_k& 0& 0& 0& \dots & 0\end{array}\right)\left(\begin{array}{c}{f}_0\\ f_1\\ \cdot \\ f_{T-k-1}\\ \cdot \\ f_{T-1}\end{array}\right).$$

$${\left(C^{T}f,g\right)}_{{\mathcal{F}}^T}={\left(u_{\cdot ,T}^f,u_{\cdot ,T}^g\right)}_{{\mathcal{H}}^T}={\left(W^{T}f,W^{T}g\right)}_{{\mathcal{H}}^T}.$$

$$C^T=C_{ij}^T,C_{ij}^T=\sum _{k=0}^{T-\max i,j}{r}_{|i-j|+2k},r_0=a_0=1.$$

$$C^T=\left(\begin{array}{ccccc}{r}_0+r_2+\dots +r_{2T-2}& r_1+\dots +r_{2T-3}& \dots & r_T+r_{T-2}& r_{T-1}\\ r_1+r_3+\dots +r_{2T-3}& r_0+\dots +r_{2T-4}& \dots & \dots & r_{T-2}\\ \cdot & \cdot & \cdot & \cdot & \cdot \\ r_{T-3}+r_{T-1}+r_{T+1}& \dots & r_0+r_2+r_4& r_1+r_3& r_2\\ r_T+r_{T-2}& \dots & r_1+r_3& r_0+r_2& r_1\\ r_{T-1}& r_{T-2}& \dots & r_1& r_0\end{array}\right).$$

$$\left\{\begin{array}[]{l}C^{T}_{i,j+1}+C^{T}_{i,j-1}-C^{T}_{i+1,j}-C^{T}_{i-1,j}=0,\\ C^{T}_{i,T}=r_{T-i},\,\,C^{T}_{T,j}=r_{T-j},\,\,r_{0}=1.\end{array}\right.$$

Full text

Abstract. We consider the Hamburger, Stieltjes and Hausdorff moment problems, that are problems of the construction of a Borel measure supported on a real line, on a half-line or on an interval $(0,1)$, from a prescribed set of moments. We propose a unified approach to these three problems based on using the auxiliary dynamical system with the discrete time associated with a semi-infinite Jacobi matrix. It is show that the set of moments determines the inverse dynamic data for such a system. Using the ideas of the Boundary Control method for every $N\in\mathbb{N}$ we can recover the spectral measure of a $N\times N$ block of Jacobi matrix, which is a solution to a truncated moment problem. This problem is reduced to the finite-dimensional generalized spectral problem, whose matrices are constructed from moments and are connected with well-known Hankel matrices by simple formulas. Thus the results on existence of solutions to Hamburger, Stieltjes and Hausdorff moment problems are naturally given in terms of these matrices. We also obtain results on uniqueness of the solution of moment problems, where as a main tool we use the Krein-type equations of inverse problem.

1. Introduction.

Given a sequence of numbers $s_{0},s_{1},s_{2},\ldots$ which are called moments, the classical moment problem consists in finding a Borel measure $\rho$ such that

[TABLE]

When $\operatorname{supp}{\rho}=\mathbb{R}$ the problem is called Hamburger moment problem, when $\operatorname{supp}{\rho}=(0,+\infty)$ the problem is called the Stieltjes moment problem, and when $\operatorname{supp}{\rho}=(0,1)$ the problem is called Hausforff moment problem. These three problems have received a lot of attention in the lust century, to mention [4, 21, 11, 12, 20] and references therein. In the present paper we offered an unified approach to these three classical problems bases on considering an auxiliary dynamical system with discrete time for Jacobi matrix [13, 16, 18] and ideas of the Boundary Control (BC) method [5, 7] of solving the inverse dynamic problems for hyperbolic dynamical systems.

In the second section we consider initial-boundary value problems for dynamical systems with discrete time associated with semi-infinite and finite Jacobi matrices. Following [13, 15, 16] we derive a dynamic and spectral representations of their systems. We introduce the operators of the BC method and show that the response operator, i.e the discrete analog of a dynamic Dirichlet-to-Neumann map for these systems (operators of this type are used as inverse data in dynamic inverse problems [5, 7]) has a form of convolution. The kernel of the response operator, which is called response vector, admits a spectral representation in terms of a spectral measure of corresponding Jacobi matrix. This fact establishes the relationship between spectral (measure) and dynamic (response vector) data and gives a possibility to apply some ideas of the BC method [6, 3] to solving the truncated moment problem.

In the third section we solve the truncated moment problem by extracting spectral data (i.e. the spectral measure of $N\times N$ block of Jacobi matrix) from the response vector. The main results are given in Theorems 3 and 4, which say that the solution to a truncated moment problem can be constructed by solving special finite dimensional generalized spectral problem, in which the matrices are connected with classical Hankel matrices (see [1, 20]) constructed from moments by simple transformation. Then the results on the existence of solution to all three moment problems are given in terms of inequalities for these matrices. Note that classical results for Hausdorff moment problem [11, 12] are given in completely different terms.

In the last section we obtain the results on uniqueness of the solution to Hamburger, Stieltjes and Hausdorff moment problems. The main tools in our considerations are classical Weyl-type results on the index of Jacobi matrix [1, 20] and Krein equations of inverse problem in dynamic form. For continuous systems such equations were derived firstly in [9] and in the framework of the BC method in [2, 8]; for the discrete systems they were derived in [13, 15, 16]. We also compare the results on existence for Hausdorff moment problem obtained in the paper with classical results of Hausdorff [11, 12, 21].

2. Dynamical systems with discrete time associated with Jacobi matrix. Operators of the BC method.

In this section we outline some results on forward problems for dynamical system with discrete time associated with finite and semi-infinite Jacobi matrices obtained in [13, 15, 16].

2.1. Finite Jacobi matrices

For a given sequence of positive numbers $\{a_{0},$ $a_{1},\ldots\}$ (in what follows we assume $a_{0}=1$) and real numbers $\{b_{1},b_{2},\ldots\}$, we denote by $A$ a semi-infinite Jacobi matrix

[TABLE]

For $N\in\mathbb{N}$, by $A^{N}$ we denote the $N\times N$ Jacobi matrix which is a block of (2.1) consisting of the intersection of first $N$ columns with first $N$ rows of $A$.

Introduce the notation $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$, and consider a dynamical system with discrete time associated with a finite Jacobi matrix $A^{N}$:

[TABLE]

by an analogy with continuous problems [5, 2, 8], we treat the real sequence $f=(f_{0},f_{1},\ldots)$ as a boundary control. Fixing a positive integer $T$ we denote by $\mathcal{F}^{T}$ the outer space of the system (2.2): $\mathcal{F}^{T}:=\mathbb{R}^{T}$, $f\in\mathcal{F}^{T}$, $f=(f_{0},\ldots,f_{T-1})$, we use the notation $\mathcal{F}^{\infty}=\mathbb{R}^{\infty}$ when control acts for all $t\geqslant 1$. The solution to (2.2) is denoted by $v^{f}$. Note that (2.2) is a discrete analog of an initial boundary value problem for a wave equation with a potential on an interval with the Dirichlet control at the left end and Dirichlet condition at the right end. The operator corresponding to a finite Jacobi matrix $A^{N}$ and Dirichlet condition at $n=N+1$ we also denote by $A^{N}:\mathbb{R}^{N}\mapsto\mathbb{R}^{N}$:

[TABLE]

Denote by $\phi_{n}(\lambda)$ a solution to the Cauchy problem for the following difference equation

[TABLE]

Thus $\phi_{k}(\lambda)$ is a polynomial of degree $k-1$. Denote by $\{\lambda_{k}\}_{k=1}^{N}$ the roots of the equation $\phi_{N+1}(\lambda)=0$, it is known [1, 20] that they are real and distinct. We introduce the vectors $\phi^{n}\in\mathbb{R}^{N}$ by the rule $\phi^{n}_{i}:=\phi_{i}(\lambda_{n})$, $n,i=1,\ldots,N,$ and define the numbers $\rho_{k}$ by

[TABLE]

where $(\cdot,\cdot)$ is a scalar product in $\mathbb{R}^{N}$.

Definition 1**.**

The set of pairs

[TABLE]

is called Dirichlet spectral data of operator $A^{N}$.

Definition 2**.**

For $f,g\in\mathcal{F}^{\infty}$ we define the convolution $c=f*g\in\mathcal{F}^{\infty}$ by the formula

[TABLE]

Denote by $\mathcal{T}_{k}(2\lambda)$ the Chebyshev polynomials of the second kind: they are obtained as a solution to the following Cauchy problem:

[TABLE]

In [13, 16] the following expression for the solution $v^{f}$ was proved:

Proposition 1**.**

The solution to (2.2) admits a representation

[TABLE]

The inner space of dynamical system (2.2) is denoted by $\mathcal{H}^{N}:=\mathbb{R}^{N}$, $h\in\mathcal{H}^{T}$, $h=(h_{1},\ldots,h_{N})$. By (2.5) we have that $v^{f}_{\cdot,\,T}\in\mathcal{H}^{N}$. For the system (2.2) the control operator $W^{T}_{N}:\mathcal{F}^{T}\mapsto\mathcal{H}^{N}$ is defined by the rule

[TABLE]

The input $\longmapsto$ output correspondence in the system (2.2) is realized by a response operator: $R^{T}:\mathcal{F}^{T}\mapsto\mathbb{R}^{T}$, defined by the formula

[TABLE]

This operator has a form of a convolution:

[TABLE]

where the convolution kernel is called a response vector: $r^{N}=(r^{N}_{0},r^{N}_{1},\ldots,r^{N}_{T-1})$. The response operator plays the role of dynamic inverse data [5, 7], for the discrete systems see [13, 16].

The connecting operator $C^{T}_{N}:\mathcal{F}^{T}\mapsto\mathcal{F}^{T}$ for the system (2.2) is defined via the quadratic form: for arbitrary $f,g\in\mathcal{F}^{T}$ one has that

[TABLE]

The speed of a wave propagation in the system (2.2) is finite, which implies the following dependence of inverse data on coefficients $\{a_{n},b_{n}\}$: for $M\in\mathbb{N}$, $M\leqslant N,$ the element $v^{f}_{1,2M-1}$ depends on $\left\{a_{1},\ldots,a_{M-1}\right\},$ $\left\{b_{1},\ldots,b_{M}\right\}$, on observing this we can formulate the following

Remark 1**.**

The entries of the response vector $(r_{0}^{N},r_{1}^{N},\ldots,r_{2N-2}^{N})$) depends on $\{a_{0},\ldots,a_{N-1}\}$, $\{b_{1},\ldots,b_{N}\}$, and does not depend on the boundary condition at $n=N+1$, the entries starting from $r_{2N-1}^{N}$ does ”feel” the boundary condition at $n=N+1$.

On introducing the special control $\delta=(1,0,0,\ldots)$, one can see that the kernel of the response operator (2.6) is given by

[TABLE]

The spectral function of operator $A^{N}$ is introduced by the rule

[TABLE]

then from (2.5), (2.7) we immediately deduce

Proposition 2**.**

The solution to (2.2), the response vector of (2.2) and entries of the matrix of the connecting operator $C^{T}_{N}$ admit the following spectral representations:

[TABLE]

2.2. Semi-infinite Jacobi matrix

We consider an initial boundary value problem for a dynamical system with discrete time associated with a semi-infinite Jacobi matrix $A$:

[TABLE]

which is a discrete analog of an initial boundary value problem for a wave equation with a potential on a half-line with the Dirichlet control at zero. The solution to (2.12) is denoted by $u^{f}_{n,\,t}$. We fix some positive integer $T$ and denote by $\mathcal{F}^{T}$ the outer space of the system (2.12), the space of controls (inputs): $\mathcal{F}^{T}:=\mathbb{R}^{T}$, $f\in\mathcal{F}^{T}$, $f=(f_{0},\ldots,f_{T-1})$.

Lemma 1**.**

A solution to (2.12) admits the representation

[TABLE]

*where $w_{n,s}$ satisfies certain Goursat problem. *

The input $\longmapsto$ output correspondence in the system (2.12) is realized by a response operator: $R^{T}:\mathcal{F}^{T}\mapsto\mathbb{R}^{T}$ defined by the rule

[TABLE]

This operator plays the role of inverse data, the corresponding inverse problem was considered in [13, 16]. The convolution kernel of $R^{T}$ is called a response vector, in accordance with (2.13) one has that $r=(r_{0},r_{1},\ldots,r_{T-1})=(1,w_{1,1},w_{1,2},\ldots w_{1,T-1})$:

[TABLE]

By choosing a special control $f=\delta=(1,0,0,\ldots)$, the kernel of the response operator can be determined as

[TABLE]

For a fixed $T\in\mathbb{N}$ we introduce the inner space of the dynamical system (2.12) $\mathcal{H}^{T}:=\mathbb{R}^{T}$, $h\in\mathcal{H}^{T}$, $h=(h_{1},\ldots,h_{T})$, the space of states. The wave $u^{f}_{\cdot,\,T}$ is considered as a state of the system (2.12) at the moment $t=T$. By (2.13) we have that $u^{f}_{\cdot,\,T}\in\mathcal{H}^{T}$. The input $\longmapsto$ state correspondence of the system (2.12) is realized by a control operator $W^{T}:\mathcal{F}^{T}\mapsto\mathcal{H}^{T}$, defined by the rule

[TABLE]

From (2.13) we deduce the representation for $W^{T}$:

[TABLE]

Or in matrix form:

[TABLE]

The following statement is equivalent to a boundary controllability of (2.12):

Lemma 2**.**

The operator $W^{T}$ is an isomorphism between $\mathcal{F}^{T}$ and $\mathcal{H}^{T}$.

We introduce the connecting operator $C^{T}:\mathcal{F}^{T}\mapsto\mathcal{F}^{T}$ for the system (2.12), by the quadratic form: for arbitrary $f,g\in\mathcal{F}^{T}$ we define

[TABLE]

That is $C^{T}=\left(W^{T}\right)^{*}W^{T}$. The fact that the connecting operator can be represented in terms of inverse data is crucial in the BC method.

Theorem 1**.**

The connecting operator $C^{T}$ is an isomorphism in $\mathcal{F}^{T}$, it admits the representation in terms of dynamic inverse data:

[TABLE]

[TABLE]

One can observe [13] that $C^{T}_{ij}$ satisfies the difference boundary problem:

Corollary 1**.**

The kernel of $C^{T}$ satisfies

[TABLE]

With the matrix $A$ we associate the operator $A$ (we keep the same notation), defined on $l^{2}\ni\phi=(\phi_{1},\phi_{2},\ldots)$, and given by

[TABLE]

By $d\rho(\lambda)$ we denote the spectral measure of $A$ (non-unique if $A$ is in the limit circle case at infinity), see [1, 4].

The Remark 1 in particular implies that

[TABLE]

Thus due to (2.18), we have that $r_{t-1}=r_{t-1}^{N}$, $t=1,\ldots,2N$. On the other hand, taking into the account (2.19), we can see that $C^{T}=C^{T}_{N}$ with $T\leqslant N$. Thus, taking in (2.10), (2.11) $N\to\infty$, we obtain the

Proposition 3**.**

The entries of the response vector of (2.12) and of the matrix of the connecting operator $C^{T}$ admit the spectral representation:

[TABLE]

3. Truncated moment problem. Recovering ”Dirichlet” spectral data from dynamic data.

We make the following observation: in the moment problem one recovers the measure from the given set of moments (1.1), in the dynamic inverse problem [13, 16] one recovers the Jacobi matrix from the given response vector (2.14). The spectral representation of response vector (2.20),the results from the previous section and from [13, 16] implies the following

Remark 2**.**

The knowledge of the finite set of moments $\{s_{0},s_{1},\ldots,s_{2N-2}\}$ is equivalent to the knowledge of $\{r_{0},r_{1},\ldots,r_{2N-2}\}$, wrom which it is possible to recover Jacobi matrix $A^{N}\in\mathbb{R}^{N\times N}$ whose elements can be thought of as a coefficients in dynamical system (2.2) with Dirichlet boundary condition at $n=N+1$, or $N\times N$ block in semi-infinite Jacobi matrix in (2.2) with no condition at the right end.

Definition 3**.**

By a solution of a truncated moment problem of order $N$ we call a Borel measure $d\rho(\lambda)$ on $\mathbb{R}$ such that equalities (1.1) with this measure hold for $k=0,1,\ldots,2N-2.$

In [16] the authors proved the following

Theorem 2**.**

The vector $(r_{0},r_{1},r_{2},\ldots,r_{2N-2})$ is a response vector for the dynamical system (2.2) if and only if the matrix $C^{N}$ defined by (2.21), (2.17) is positive definite.

This theorem and formulas for the entries of Jacobi matrix, which will be provided later, imply the following procedure of solving the truncated moment problem:

Calculate $(r_{0},r_{1},r_{2},\ldots,r_{2N-2})$ from $\{s_{0},s_{1},\ldots,s_{2N-2}\}$ by using (2.20).

• 2)

Recover $N\times N$ Jacobi matrix $A^{N}$ using formulas for $a_{k},$ $b_{k}$ from [16]

• 3)

Recover spectral measure for finite Jacobi matrix $A^{N}$ with prescribed arbitrary selfadjoint condition at $n=N+1$.

• 3’)

Extend Jacobi matrix $A^{N}$ to finite Jacobi matrix $A^{M}$, $M>N$, prescribe arbitrary selfadjoint condition at $n=M+1$ and recover spectral measure of $A^{M}$.

• 3”)

Extend Jacobi matrix $A^{N}$ to infinite Jacobi matrix $A$, and recover spectral measure of $A$.

Every measure obtained in $3),$ $3^{\prime})$, $3^{\prime\prime})$ gives a solution to the truncated moment problem. Below we propose a different approach: using the ideas of the BC method we recover the spectral measure corresponding to Jacobi matrix $A^{N}$ directly from moments (from the operator $C^{N}$), without recovering the Jacoi matrix itself.

Agreement 1**.**

We assume that controls $f\in\mathcal{F}^{N}$, $f=\left(f_{0},\ldots,f_{N-1}\right)$ are extended: $f=\left(f_{-1},f_{0},\ldots,f_{N-1},f_{N}\right)$, where $f_{-1}=f_{N}=0$.

We introduce the special space of controls $\mathcal{F}^{N}_{0}=\left\{f\in\mathcal{F}^{T}\,|\,f_{0}=0\right\}$ and the operator $D:\mathcal{F}^{N}\mapsto\mathcal{F}^{N}$ acting by

[TABLE]

The following statement can be easily proved using arguments from [16] and representations (2.15) and (2.13):

Proposition 4**.**

The operator $W^{N}$ maps $\mathcal{F}^{N}_{0}$ isomorphically onto $\mathcal{H}^{N-1}$.

• 2)

On the set $\mathcal{F}^{N}_{0}$ the following relation holds:

[TABLE]

Taking $f,g\in\mathcal{F}_{0}^{N}$ and evaluating the quadratic form, bearing in mind (3.1), we obtain:

[TABLE]

The last equality in (3.2) means that only $A^{N-1}$ block of the whole matrix $A^{N}$ is in use. Then it is possible to perform the spectral analysis of $A^{N-1}$ using the classical variational approach, the controllability of the system (2.2) (see Proposition 4) and the representation (3.2), see also [6]. The spectral data of Jacobi matrix $A^{N-1}$ with the Dirichlet boundary condition at $n=N$ can be recovered by the following procedure:

The first eigenvalue is given by

[TABLE]

• 2)

Let $f^{1}$, be the minimizer of (3.3), then

[TABLE]

• 3)

The second eigenvalue is given by

[TABLE]

• 4)

Let $f^{2}$, be the minimizer of (3.4), then

[TABLE]

Continuing this procedure, one recovers the set $\{\lambda_{k}^{N-1},\rho_{k}\}_{k=1}^{N-1}$ and constructs the measure $d\rho^{N-1}(\lambda)$ by (2.8).

Remark 3**.**

The measure, constructed by the above procedure solves the truncated moment problem for the set of moments $\{s_{0},s_{1},\ldots,s_{2N-4}\}$.

3.1. Euler-Lagrange equations

In this section we derive equations which can be thought of as a Euler-Lagrange equations for the problem of the minimization of a functional $\left(C^{N}Df,f\right)_{\mathcal{F}^{N}}$ in $\mathcal{F}^{N}_{0}$ with the constrain $\left(C^{T}f,f\right)_{\mathcal{F}^{N}}=1$, described in the previous section. Similar method of deriving equations which can be used for recovering of spectral data was used in [3].

By $f^{k}$, $k=1,\ldots,N$ we denote the control that drive system (2.2) to prescribed state (see (2.3)):

[TABLE]

Due to Proposition 4, such a control exists and is unique for every $k$. We introduce the operator

[TABLE]

and denote by $P:\mathcal{F}^{N+1}\mapsto\mathcal{F}^{N}$ the embedding operator. Then $P^{*}:\mathcal{F}^{N}\mapsto\mathcal{F}^{N+1}$ extends vector by zero: $\left(P^{*}f\right)_{k}=f_{k}$, $k=0,1,N-1$, and $\left(P^{*}f\right)_{N}=0.$

Theorem 3**.**

The spectrum of $A^{N}$ and (non-normalized) controls $f_{k}$, $k=1,\ldots,N$ are the spectrum and the eigenvectors of the following generalized spectral problem:

[TABLE]

Proof.

For $h\in\mathcal{F}^{T}$ we always assume that $h_{-1}=h_{T}=0$ (see Agreement 1). For a fixed $k=1,\ldots,N$ we take $f_{k}\in\mathcal{F}^{N}$ such that $W^{N}f_{k}=v^{f_{k}}_{\cdot,\,N}=\phi^{k}$, then for arbitrary $g\in\mathcal{F}^{N}$ we can evaluate:

[TABLE]

We note that

[TABLE]

That is why we can rewrite the first summand in the right hand side of (3.6) as

[TABLE]

Analogously:

[TABLE]

So we can rewrite the second summand in the right hand side of (3.6) as

[TABLE]

Finally from (3.6), (3.7) and (3.8) we deduce that

[TABLE]

Using operators $P,$ $P^{*}$ we can rewrite (3.9) in the form

[TABLE]

Thus the pair $\{f_{k},\lambda_{k}\}$ gives the solution to (3.5). Now let the pair $\{f,\lambda\}$ be the solution to (3.5) with $f\in\mathcal{F}^{N}$ $f\not=f_{k},$ $\lambda\not=\lambda_{k}$ for any $k=1\,\ldots,N$. Then $W^{N}f=v^{f}_{\cdot,\,N}=\sum_{k=1}^{N}a_{k}\phi^{k}$ for some $a_{k}\in\mathbb{R}$. We can evaluate for arbitrary $g\in\mathcal{F}^{N}$:

[TABLE]

From the above equality and Proposition 4 it follows that all $a_{k}$ except one are equal to zero, and for such $a_{j},$ $\lambda=\lambda_{j}$, which completes the proof. ∎

Having found spectrum and non-nomalized controls from (3.5) we can recover the measure of operator $A^{N}$ with Dirichlet boundary condition at $n=N+1$ by the following procedure:

Normalize controls by choosing $\left(C^{N}f_{k},f_{k}\right)_{\mathcal{F}^{N}}=1$,

• 2)

Observe that $W^{N}f^{k}=\alpha_{k}\phi^{k}$ for some $\alpha_{k}\in\mathbb{R}$, where the constant is defined by $\alpha_{k}=\left(Rf_{k}\right)_{N}$.

• 3)

The norming coefficients are given by $\rho_{k}={\alpha_{k}^{2}}$, $k=1,\ldots,N$.

• 4)

Recover the measure by (2.8).

Now we rewrite the generalized spectral problem (3.5) in more details and transfer the matrices in left and right-hand sides to Hankel matrices known from classical literature [1, 20]. Note that the matrices in (3.5) has the following representations:

[TABLE]

Here we used the notations for entries of $C^{N}$ different from ones in (2.17) in order to show that $C^{N}$ is a lower right block in $C^{N+1}$. The left hand side of (3.5) we denote by

[TABLE]

Proposition 5**.**

The matrix $B^{N}$ is self-adjoint, it admits the following representation:

[TABLE]

Proof.

We note that the matrices $P$, $P^{*}$ and $D_{-}$, $D^{*}_{-}$ have one diagonal filled with ones and the other elements are zeros. Thus the multiplication by such a matrix leads to deleting a line or column from the original matrix (possibly with the addition of a zero line or column). Performing calculations we see that the first term in the right hand side of $B^{N}$ is obtained by deleting last column and first row from $C^{N+1}$ and the second term is obtained by deleting the first column and adding zero column to $C^{N}$. All aforesaid leads to the formula (3.11).

We note that the above representation (3.11) and Corollary 1 shows that $B^{T}$ is self-adjoint matrix. ∎

Remark 4**.**

The spectral problem (3.5) has a form

[TABLE]

Chebyshev polynomials of the second kind $\{\mathcal{T}_{1}(\lambda),\mathcal{T}_{2}(\lambda),\ldots\mathcal{T}_{n}(\lambda)\}$ (see 2.4) are related to $\{1,\lambda,\lambda^{n-1}\}$ by the following relation

[TABLE]

Proposition 6**.**

The entries of the matrix $\Lambda_{n}\in R^{n-1\times n-1}$ are given by

[TABLE]

The entries of the response vector are related to moments by the rule:

[TABLE]

Proof.

The formula (3.14) for entries of $\Lambda_{n}$ is proved by direct calculations with the use of properties of Chebyshev polynomials. Then making use of (2.20) yields (3.15). ∎

Introduce the following Hankel matrices

[TABLE]

the matrix $J_{N}\in\mathbb{R}^{N\times N}$:

[TABLE]

and define

[TABLE]

The remarkable fact is that the matrices $B^{N},\,C^{N}$ can be reduced to Hankel matrices by the same linear transformation:

Theorem 4**.**

The following relations hold:

[TABLE]

Then the generalized spectral problem (3.5) or (3.12) upon introducing the notation $g_{k}=\left(\widetilde{\Lambda}_{N}\right)^{*}f_{k}$ is equivalent to the following generalized spectral problem:

[TABLE]

Proof.

Using (3.10) and the representation (2.21), we have that entries of $C^{N}$ have a form:

[TABLE]

We can write down the form of the operator $C^{N}$ then:

[TABLE]

Using (3.13) we can rewrite (3.19) as

[TABLE]

which proves (3.16).

Using the representation of $B^{N}$ (3.11) and (2.21) yields the following formula for entries $b_{ij}$ of $B^{N}$:

[TABLE]

where we counted that $\mathcal{T}_{0}(\lambda)=0$. Making use of (2.4) leads to:

[TABLE]

Then using (3.13) and (3.20) we obtain:

[TABLE]

which gives (3.17). Then (3.18) is a consequence of (3.16) and (3.17). ∎

3.2. Special cases: Hamburger, Stieltjes and Hausdorff moment problems

In the previous sections we constructed the special measure corresponding to operator $A^{N}$ with Dirichlet boundary condition at $n=N+1$ and which gives a solution of a truncated moment problem. Here we formulate several consequences of Theorems 3, 4.

Bearing in mind the relationship between elements of response vector and moments (2.20) and the formula (3.16), we can reformulate Theorem 2 as

Proposition 7**.**

The set of numbers $(s_{0},s_{1},s_{2},\ldots,s_{2N-2})$ are moments of a spectral measure corresponding to the Jacobi operator $A^{N}$ with Dirichlet boundary condition at $n=N+1$ if and only if

[TABLE]

The Stieltjes moment problem is characterized by the positivity of a support of a measure. That means the positivity of a spectrum of $A^{N}$. The latter leads to the following

Proposition 8**.**

The set of numbers $(s_{0},s_{1},s_{2},\ldots,s_{2N-1})$ are moments of a spectral measure, supported on $(0,+\infty)$, corresponding to Jacobi operator $A^{N}$ with Dirichlet boundary condition at $n=N+1$ if and only if

[TABLE]

In the Hausdorff moment problem the measure is supported on $(0,1)$, which leads to the following

Proposition 9**.**

The set of numbers $(s_{0},s_{1},s_{2},\ldots,s_{2N-1})$ are moments of a spectral measure, supported on $(0,1)$, corresponding to operator $A^{N}$ with Dirichlet boundary condition at $n=N+1$, if and only if the condition

[TABLE]

holds.

Proof.

From (3.18) it follows that

[TABLE]

Then the restriction $\lambda_{k}\in(0,1)$ implies (3.23). ∎

Remark 5**.**

Given an infinite sequence of moments, one can determine whether or not it is Hamburger or Stieltjes or Hausdorff moment sequence by verifying condition (3.21) or (3.22) or (3.23) holds for all $N\in\mathbb{N}$.

3.3. Recovering Jacobi matrix, nonuniqueness of the solution of the truncated moment problem.

As we mentioned, given a sequence of moments $\{s_{0},s_{1},\ldots,s_{2N-2}\}$ or, equivalently, entries of the response vector, $\{r_{0},r_{1},\ldots,r_{2N-2}\}$, it is possible to recover the Jacobi matrix $A^{N}$ [13, 16]. Introduce the matrices

[TABLE]

that is $\overline{C}^{k-1}_{k}$ is constructed from the matrix $C^{k-1}$ by substituting the last column by $(\overline{c}_{1,\,k},\ldots,\overline{c}_{k-1,\,k})^{T}$.

Proposition 10**.**

The entries of Jacobi matrix can be recovered by

[TABLE]

where we set $\det{C^{0}}=1,$ $\det{C^{-1}=1}$.

[TABLE]

Note that (3.25), (3.26) are the consequences of the relation

[TABLE]

and representation (2.15) of $W^{T}$.

Remark 6**.**

In order to apply the results of Theorems 3, 4, i.e. the generalized spectral problem (3.18) to the problem of reconstruction of spectral measure of $A^{N}$ one need to know one extra moment, specifically $s_{2N-1}$ (see the definition of $S^{N}_{1}$), than in the method based on direct calculation of $A^{N}$ by formulas (3.25) and (3.26).

Denote by $B(\mathbb{R})$ the space of Borel measures on $\mathbb{R}$ and by $M_{N}\subset B(\mathbb{R})$ a subset such that $d\nu(\lambda)\in M_{N}$ is a solution of the truncated moment problem (1.1) of the order $N$. We used the BC method to construct the special solution of a truncated moment problem: for $N\in\mathbb{N}$ the set of moments $\{s_{0},s_{1},\ldots,s_{2N-1}\}$ determines the measure $d\rho^{N}(\lambda)\in M_{N}$, where the constructed measure is a spectral measure of a finite Jacobi operator $A^{N}$ with the Dirichlet condition at $n=N+1$. We point out that in our procedure we do not use the Jacobi matrix, but rather special Hankel matrices, constructed from moments.

Having constructed the Jacobi matrix $A^{N}$ from the set $\{s_{0},s_{1},\ldots,s_{2N-2}\}$ we can consider the operator $\widetilde{A}^{N}$ given by mixed boundary condition at $n=N+1$: $\alpha\phi_{N}+\beta\phi_{N+1}=0$ for some $\alpha,\beta\in\mathbb{R}$, $|\alpha|+|\beta|>0$, or one can extend the matrix $A^{N}$ in any way, keeping it to be Jacobi (it is possible that it would be necessary to add the boundary condition at infinity). Then the spectral measure to any of described operators also gives a solution to Hamburger moment problem.

Remark 7**.**

The spectral representation of (2.21) implies that $M_{N}$ is a convex set in $B(\mathbb{R})$, and obviously $M_{N_{1}}\subseteq M_{N_{2}}$ when $N_{1}>N_{2}$. Taking $N$ to infinity we deduce that the set of solutions $M_{\infty}$ of the Hamburger moment problem (1.1) either convex, or consists of one element. The same arguments and spectral representation of $B^{T}$ (3.20) shows that the set of solutions $M_{\infty}^{s}$ to Stieltjes moment problem either convex or consists of one element.

4. On the uniqueness of the solution of the Hamburger, Stieltjes and Hausdorff moment problems.

We remind the reader that the moment problem is called determinate if it has only one solution, otherwise it is called indeterminate.

In this section we use the complex-valued outer and inner spaces for the dynamical system (2.12): $\mathcal{F}^{T}=\mathcal{H}^{T}=\mathbb{C}^{T}$ with the scalar products $(f,g)_{\mathcal{F}^{T}}=\sum_{i=0}^{T-1}f_{i}\overline{g}_{i}$ and $(a,b)_{\mathcal{H}^{T}}=\sum_{i=1}^{T}a_{i}\overline{b_{i}}$.

4.1. Krein equations

Let $\alpha,\beta\in\mathbb{R}$ and $y(\lambda)$ be a solution to a Cauchy problem for the following difference equation (we remind the agreement $a_{0}=1$):

[TABLE]

We set up the special control problem: to find a control $f^{T}\in\mathcal{F}^{T}$ that drives the system (2.12) to the prescribed state $\overline{y^{T}(\lambda)}:=\left(\overline{y_{1}(\lambda)},\ldots,\overline{y_{T}(\lambda)}\right)\in\mathcal{H}^{T}$ at $t=T$:

[TABLE]

Note that due to Lemma 2, this control problem has a unique solution $f^{T}=\left(W^{T}\right)^{-1}\overline{y^{T}(\lambda)}$. Let $\varkappa^{T}(\lambda)$ be a solution to

[TABLE]

One can easily see the relation with Chebyshev polynomials (2.4):

[TABLE]

It is an important fact that the control $f^{T}$ can be found as a solution to certain equation:

Theorem 5**.**

The control $f^{T}=\left(W^{T}\right)^{-1}\overline{y^{T}(\lambda)}$ solving the special control problem (4.2), is a unique solution to the following Krein-type equation in $\mathcal{F}^{T}$:

[TABLE]

Proof.

Let $f^{T}$ be a solution to (4.2). We observe that for any fixed $g\in\mathcal{F}^{T}$ we have that

[TABLE]

Indeed, changing the order of a summation in the right hand side of (4.6) and counting $u_{k,\,-1}^{g}=u_{k,\,0}^{g}=0$ yields

[TABLE]

which gives (4.6) due to (4.3). Using this observation, we can evaluate

[TABLE]

Which completes the proof due to the arbitrariness of $g$. ∎

We consider two special solutions to (4.1): the first one $\varphi(\lambda)$ corresponds to the choice $\alpha=0,$ $\beta=1$, the second one, $\xi(\lambda)$, corresponds to Cauchy data $\alpha=-1$, $\beta=0$.

It is well-known fact [1, 20] that the questions on the uniqueness of the solution to a moment problem are related to the index of the operator $A$. Here we provide well-known results on discrete version of Weyl limit point-circle theory which answers the question on the index of $A$ that will be subsequently used:

Proposition 11**.**

The Jacobi operator $A$ is limit circle at infinity (has index equal to one) if and only if one of the following occurs:

$\varphi(x),\xi(x)\in l^{2}$* for some $x\in\mathbb{R}$,*

• 2)

$\varphi(x),\varphi^{\prime}(x)\in l^{2}$* for some $x\in\mathbb{R}$,*

• 3)

$\xi(x),\xi^{\prime}(x)\in l^{2}$* for some $x\in\mathbb{R}$.*

4.2. Hamburger moment problem

Let in (4.1) $\alpha=0,$ $\beta=1$, then the special control problem has a form:

[TABLE]

The control $f^{T}_{01}$ is a unique solution to (see (4.5) (4.4)) the equation

[TABLE]

Differentiating (4.7), (4.8) with respect to $\lambda$, we see that

[TABLE]

and the control $\left(f^{T}_{01}(\lambda)\right)^{\prime}$ is a solution to

[TABLE]

Evaluating the quadratic form (2.16) we have that

[TABLE]

And similarly for the derivatives:

[TABLE]

It is known that

[TABLE]

We define the vectors

[TABLE]

Using the above arguments we can state that

[TABLE]

Now we can use 2) from Proposition 11, and formulate the following

Proposition 12**.**

The Hamburger moment problem is indeterminate if and only if

[TABLE]

where $\Gamma_{T}$ and $\Omega_{T}$ are defined by (4.11), (4.9), (4.10).

4.3. Stieltjes moment problem

It is known [1] that the Jacobi matrix in this case admits the special structure:

[TABLE]

where $l_{i}$, $m_{i}$ are positive and are interpreted as lengths of intervals and masses at the points $x_{j}$. The string is defined by the density $dM=\sum_{k=1}^{\infty}m_{k}\delta(x-x_{k})$, where $0=x_{0}<x_{1}<x_{2}<\ldots<x_{N-1}<\ldots$, $l_{i}=x_{i}-x_{i-1}$, $i=1,\ldots$. The inverse dynamic problem for the dynamical system corresponding to a finite Krein-Stieltjes string was studied in [17]. It is straightforward to check (see also [1]) that the following relations hold:

[TABLE]

We define the mass and length of a segment of a string:

[TABLE]

when $K=\infty$ above expressions correspond to the mass and the length of the whole string. Then formulas (4.14), (4.15) imply that

[TABLE]

The immediate consequence of 1) in Proposition 11 and formulas (4.14), (4.15) is the following

Proposition 13**.**

The Stieltjes moment problem is indeterminate if and only if both length and mass of a string is finite: $M_{\infty},\,L_{\infty}<+\infty$.

For the mass of the segment of a string or of the whole string we have an expression (4.12). Now we obtain similar formula for the length. Denote by $h^{T}\in\mathcal{F}^{T}$ the (unique) control which drives the system (2.12) to the special state

[TABLE]

For arbitrary $q\in\mathcal{F}^{T}$ we have (see the representation (2.15)) that

[TABLE]

The above relation implies that $h^{T}$ can be found as a unique solution to Krein-type equation:

[TABLE]

Taking a control $f^{T}\in\mathcal{F}^{T}$ such that $W^{T}f^{T}=\left(\varphi_{1}(0),\ldots,\varphi_{T}(0)\right)$ we have (see Theorem 5 and (4.16) that

[TABLE]

Similarly, denote by $g^{T}$ the control for which $W^{T}g^{T}=\left(\xi_{1}(0),\ldots,\xi_{T}(0)\right)$. Then, using equations from Theorem 5 and (4.16) we have that

[TABLE]

The above arguments leads to the following expression for the length of the segment of the string:

[TABLE]

where we denoted $e_{1}=(1,0,\ldots,0)\in\mathcal{F}^{T}$. The above arguments leads to the following statement:

Proposition 14**.**

The Stieltjes moment problem is indeterminate if and only if the following relations hold:

[TABLE]

4.4. Hausdorff moment problem

It is a special case of a Stieltjes moment problem. The necessary and sufficient conditions for a set of numbers to be a moments of a measure supported on $(0,1)$ are obtained in [H], see also [21]. They are equivalent to inequality (3.23) holds for all $T\in\mathbb{N}$ since we get the limit measure as a limit of measures supported on $(0,1)$.

Proposition 15**.**

If the Halkel matrices $S_{0}^{T}$, $S_{1}^{T}$ satisfy (3.23) for all $T\geqslant 2$, then there exists only one measure supported on $(0,1)$ which satisfies (1.1). In other words, the Hausdorff moment problem is determinate.

Proof.

Let us assume that the opposite is true and the Hausdorff moment problem is indeterminate, in this case by Proposition 14 the length and the mass of string determined by the matrix $A$ constructed from moments, should be finite. Then for any fixed $T\in\mathbb{N}$ we have on the one hand that

[TABLE]

On the other hand (see (4.13)),

[TABLE]

Since $l_{i},\,m_{i}$ are positive and by our assumption $\sum_{i=1}^{\infty}m_{i}<+\infty,\,\,\sum_{i=1}^{\infty}l_{i}<+\infty$, it immediately follows from (4.18) that $\operatorname{tr}A^{T}>T^{2}$ for sufficiently large $T$, and thus from (4.17) we can see that for such $T$ the eigenvalues $\lambda^{T}_{k}$ cannot be bounded by one. Which gives a contradiction. ∎

Let $s\in\mathbb{R}^{\mathbb{N}}$ be a sequence, $s=\left(s_{0},s_{1},\ldots\right)$. One defines the difference operator $\Delta:\mathbb{R}^{\mathbb{N}}\mapsto\mathbb{R}^{\mathbb{N}}$ by the rule

[TABLE]

Hausdorff in [11, 12] proved the following

Theorem 6**.**

A sequence $s=\left(s_{0},s_{1},\ldots\right)\in\mathbb{R}^{\mathbb{N}}$ is a moment sequence of a measure supported on $(0,1)$ if and only if it is completely monotonic, i.e., its difference sequences satisfy the equalities

[TABLE]

We will show that the Hausdorff condition (4.20) is a consequence of condition (3.23) holding far all $N\in\mathbb{N}$. On the other hand, the property (4.20) for finite $k,n$ does not imply (3.23). The following two propositions confirm that.

Proposition 16**.**

The condition (3.23) implies the inequality (4.20) holds for $k,\,n$ such that $k+n\leqslant 2N-1$.

Proof.

According to definition (4.19)

[TABLE]

continuing calculations yields

[TABLE]

For a given sequence $g=\left(g_{0},g_{1},\ldots\right)\in\mathbb{R}^{\mathbb{N}}$ by $\left(g_{i+j-2}\right)_{1\leq i,j\leq n+1}$ we denote the Hankel matrix, and the Hankel transform get map $g\in\mathbb{R}^{\mathbb{N}}$ to the sequence

[TABLE]

The binomial transform get map a given sequence $g=\left(g_{0},g_{1},\ldots\right)\in\mathbb{R}^{\mathbb{N}}$ to the sequence

[TABLE]

It is known [19] that the Hankel transform is invariant under the binomial transform, that is:

[TABLE]

Note that (4.23) remains valid if one replaces binomial transform (4.22) by the signed binomial transform introduced by the rule: for $g=\left(g_{0},g_{1},\ldots\right)\in\mathbb{R}^{\mathbb{N}}$ one has

[TABLE]

This fact and (4.21) imply that for Hankel matrices $J_{N}S^{N}_{0}J_{N}=\left(s_{i+j-2}\right)_{1\leq i,j\leq N}$, $J_{N}S^{N}_{1}J_{N}=\left(s_{i+j-1}\right)_{1\leq i,j\leq N}$ and their difference the following relations hold:

[TABLE]

where $n=0,1,\ldots,N-1$. Then the Sylvester criterion of positivity of a matrix and the condition (3.23) imply that

[TABLE]

The above inequalities imply that diagonal elements of matrices satisfy the same inequalities

[TABLE]

Using the definition of $\Delta$ (4.19) from (4.25) we derive that

[TABLE]

Note that inequalities in (4.25),(4.26) are only part of what we need to show in (4.20). To prove the other part we note that condition (3.23) implies that

[TABLE]

Like (4.25) was a consequence of (3.23), the inequality

[TABLE]

follows from (4.27). From (4.28) one obtains that

[TABLE]

The inequalities (4.28), (4.29) is exactly what we need to show in (4.20). ∎

Proposition 17**.**

The inequality (4.20) holding for all $k,n:\ k+n\leqslant 2N-1$ does not implies (3.23).

Proof.

Consider the point mass measure concentrated at $\lambda=1/2$: i.e. $d\rho(\lambda)=\delta(\lambda-1/2)$. In this case all moments (1.1) are given by $s_{k}=(1/2)^{k}$. Using (4.19) we see that $\left(\Delta^{k}s\right)_{n}=(-1)^{k}(1/2)^{k+n}$, which agrees with (4.20).

Now we construct counterexample which works even for $N=2$. To do so we slightly change the moment $s_{2}$: we take $s_{0}=1$, $s_{1}=1/2$, $s_{2}=1/4+\varepsilon$, $s_{3}=1/8$. If $\varepsilon$ is small enough then (4.20) remains valid, but (3.23) fails: indeed, $\det(s_{i+j-2})_{1\leq i,j\leq 2}=\varepsilon<0$ if $\varepsilon<0$. Therefore (3.23) doesn’t follow from (4.20). ∎

Note that the above counterexample does not work in the case of infinite matrices: in this case the restriction on indices $k+n\leqslant 2N-1$ disappears and one should consider all $k,n$. Then according to (4.21)

[TABLE]

and we see that the second term in the right hand side of the above expression dominates if $k,n$ are large enough. Therefore for such a moment sequence the condition (4.20) does not hold.

Acknowledgments

The research of Victor Mikhaylov was supported in part by RFBR 17-01-00529, RFBR 18-01-00269. Alexandr Mikhaylov was supported by RFBR 17-01-00099 and RFBR 18-01-00269.