37 research outputs found
Inverse problems for periodic generalized Jacobi matrices
Some inverse problems for semi-infinite periodic generalized Jacobi matrices
are considered. In particular, a generalization of the Abel criterion is
presented. The approach is based on the fact that the solvability of the
Pell-Abel equation is equivalent to the existence of a certainly normalized
-unitary -matrix polynomial (the monodromy matrix).Comment: 11 pages (some typos are corrected
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 Pad\'e approximants.Comment: 17 page
The linear pencil approach to rational interpolation
It is possible to generalize the fruitful interaction between (real or
complex) Jacobi matrices, orthogonal polynomials and Pade approximants at
infinity by considering rational interpolants, (bi-)orthogonal rational
functions and linear pencils zB-A of two tridiagonal matrices A, B, following
Spiridonov and Zhedanov.
In the present paper, beside revisiting the underlying generalized Favard
theorem, we suggest a new criterion for the resolvent set of this linear pencil
in terms of the underlying associated rational functions. This enables us to
generalize several convergence results for Pade approximants in terms of
complex Jacobi matrices to the more general case of convergence of rational
interpolants in terms of the linear pencil. We also study generalizations of
the Darboux transformations and the link to biorthogonal rational functions.
Finally, for a Markov function and for pairwise conjugate interpolation points
tending to infinity, we compute explicitly the spectrum and the numerical range
of the underlying linear pencil.Comment: 22 page