Compact Jacobi matrices: from Stieltjes to Krein and M(a,b)

Generalized Jacobi operators in Krein spaces

A special class of generalized Jacobi operators which are self-adjoint in Krein spaces is presented. A description of the resolvent set of such operators in terms of solutions of the corresponding recurrence relations is given. In particular, special attention is paid to the periodic generalized Jacobi operators. Finally, the spectral properties of generalized Jacobi operators are applied to prove convergence results for Padé approximants.

Heine, Hilbert, Pade, Riemann, and Stieltjes: a John Nuttall's work 25 years later

In 1986 J. Nuttall published in Constructive Approximation the paper "Asymptotics of generalized Jacobi polynomials", where with his usual insight he studied the behavior of the denominators ("generalized Jacobi polynomials") and the remainders of the Pade approximants to a special class of algebraic functions with 3 branch points. 25 years later we try to look at this problem from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the o…

New explicitly diagonalizable Hankel matrices related to the Stieltjes-Carlitz polynomials

Four new examples of explicitly diagonalizable Hankel matrices depending on a parameter $k\in(0,1)$ are presented. The Hankel matrices are regarded as matrix operators on the Hilbert space $\ell^{2}(\mathbb{N}_{0})$ and the solution of the spectral problem is based on an application of the commutator method. Each of the Hankel matrices commutes with a Jacobi matrix which is related to a particular family of the Stieltjes-Carlitz polynomials. More examples of explicitly diagonalizable structured matrix operators are obtained when taking into account also weighted Hankel matrices.

On bounded complex Jacobi matrices and related moment problems

In this paper we study the linear functional $S$ on complex polynomials which is associated to a bounded complex Jacobi matrix $J$. The associated moment problem is considered: find a positive Borel measure $μ$ on $\mathbb{C}$ subject to conditions $\int z^n dμ= s_n$, where $s_n$ are prescribed complex numbers (moments). This moment problem may be viewed as an extension of the Stieltjes and Hamburger moment problems to the complex plane. Sufficient conditions for the solvability of the moment problem are provided. As a corollary, we obtain conditions for the existence of an integral representa…

On the relation between Darboux transformations and polynomial mappings

Let dμ(t) be a probability measure on [0,+\infty) such that its moments are finite. Then the Cauchy-Stieltjes transform S of dμ(t) is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. In the present paper, we consider a matrix interpretation of the unwrapping transformation from S(λ) to λS(λ^2), which is intimately related to the simplest case of polynomial mappings. More precisely, it is shown that this transformation is essentially a Darboux transformation of the underlying Jacobi matrix. Moreover, in this scheme, the Chihara construction of solutions to th…

来自双对角分解的矩阵连分数

Positive Bidiagonal Factorizations: Combinatorial Structure, Spectral Theory, and Matrix Continued Fractions

本文从有序双对角分解构造矩阵连分数，通过循环线性化和有理高度排序转化为块三对角形式，截断得到 Padé 型逼近，并给出收敛性、残差估计及多种谱实现。