We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
An operator approach to the indefinite Stieltjes moment problem.
- Authors
Derkach, Vladimir; Kovalyov, Ivan
- Abstract
A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class N ( κ ϵ ℤ), if f is symmetric with respect to ℝ and the kernel $$ {\mathbf{N}}_{\omega }(z)\coloneq \frac{f(z)-\overline{f\left(\omega \right)}}{z-\overline{\omega}} $$ has κ negative squares on ℂ. The generalized Stieltjes class $$ {\mathbf{N}}_{\kappa}^k\left(\kappa, k\in {\mathrm{\mathbb{Z}}}_{+}\right) $$ is defined as the set of functions f ϵ N such that z f ϵ N . The full indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ consists in the following: Given κ, k ϵ ℤ, and a sequence $$ \mathbf{s}={\left\{{s}_i\right\}}_{i=0}^{\infty } $$ of real numbers, to describe the set of functions $$ f\in {\mathbf{N}}_{\kappa}^k $$ , which satisfy the asymptotic expansion for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A generated by $$ {\mathfrak{J}}_{\left[0;N\right]} $$ . The u-resolvent matrices of the operator A are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple's technique. Some criteria for the problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ to be solvable and indeterminate are found. Explicit formulae for Padé approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.
- Subjects
MOMENT problems (Mathematics); DIOPHANTINE equations; STIELTJES transform; MATHEMATICAL symmetry; MEROMORPHIC functions
- Publication
Journal of Mathematical Sciences, 2017, Vol 227, Issue 1, p33
- ISSN
1072-3374
- Publication type
Article
- DOI
10.1007/s10958-017-3573-3