Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

Mathematische Nachrichten

Hassi

Snoo

2012

Truncated moment and related interpolation problems in the class of generalized Nevanlinna functions are investigated. General solvability criteria will be established and complete parametrizations of solutions are given. The framework used in the paper allows the treatment of even and odd order problems in a parallel manner. The main new results concern the case where the corresponding Hankel matrix of moments is degenerate. One of the new effects in the indefinite case is that the degenerate moment problem may have infinitely many solutions. However, with a careful application of an indefinite analogue of a step-by-step Schur algorithm a complete description of the set of solutions will be obtained.

Section: Schur Algorithmmentioning

confidence: 99%

Truncated moment problems in the class of generalized Nevanlinna functions

Mathematische Nachrichten

Hassi

Snoo

2012

“…In [3] it was shown that each J-unitary polynomial matrix on R admits a factorization of the form (3.90) with somewhat different elementary factors W j (λ). In [3] it was shown that each J-unitary polynomial matrix on R admits a factorization of the form (3.90) with somewhat different elementary factors W j (λ).…”

Section: Corollary 324 Every Polynomial Matrix Satisfying Conditionmentioning

confidence: 99%

On the convergence of Padé approximations for generalized Nevanlinna functions

Derevyagin

Trans. Moscow Math. Soc.

2007

We study a stepwise algorithm for solving the indefinite truncated moment problem and obtain the factorization of the matrix describing the solution of this problem into elementary factors. We consider the generalized Jacobi matrix corresponding to Magnus' continuous P -fraction that appears in this algorithm and the polynomials of the first and second kind that are solutions of the corresponding difference equation. Weyl functions and the resolution matrices for finite and infinite Jacobi matrices are computed in terms of these polynomials. Convergence of diagonal and paradiagonal Padé approximation for functions from the generalized Nevanlinna class is studied.

“…Given

ℓ, κ, k \in {double-struckZ}_{+}

, and a sequence

s = {({}, s_{j})}_{j = 0}^{ℓ}

of real numbers, describe the set

{scriptM}_{κ}^{k} false(bolds false)

of functions

f \in {boldN}_{κ}^{k}

, which satisfy the asymptotic expansion

f false(z false) = - \frac{s_{0}}{z} - \dots - \frac{s_{ℓ}}{z^{ℓ + 1}} + o (\frac{1}{z^{ℓ + 1}}) false(z = i y, y ↑ \infty false) .

Such a moment problem is called even or odd regarding to the oddness of the number

ℓ + 1

of given moments. To study this problem we use the Schur algorithm, which was elaborated in , and for the class

N_{κ}

. Applications of the Schur algorithm to degenerate moment problem in the class

N_{κ}

were given in .…”

Section: Introductionmentioning

confidence: 99%

“…To study this problem we use the Schur algorithm, which was elaborated in [5], [6] and [2] for the class N κ . Applications of the Schur algorithm to degenerate moment problem in the class N κ were given in [10].…”

Section: Introductionmentioning

confidence: 99%

The Schur algorithm for an indefinite Stieltjes moment problem

Mathematische Nachrichten

Kovalyov

2016

The nondegenerate truncated indefinite Stieltjes moment problem in the class Nκk of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the Schur step‐by‐step algorithm, which leads to the expansion of these solutions in generalized Stieltjes continuous fractions studied recently in [11]. Explicit formula for the resolvent matrix in terms of generalized Stieltjes polynomials is found.