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Recursiveness in Matrix Rational Interpolation Problems
 J. Comput. Appl. Math
, 1996
"... We consider the problem of computing solutions to a variety of matrix rational interpolation problems. These include the partial realization problem for matrix power series and NewtonPad'e, HermitePad'e, Simultaneous Pad'e, MPad'e and multipoint Pad'e approximation problems along with their ma ..."
Abstract

Cited by 18 (11 self)
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We consider the problem of computing solutions to a variety of matrix rational interpolation problems. These include the partial realization problem for matrix power series and NewtonPad'e, HermitePad'e, Simultaneous Pad'e, MPad'e and multipoint Pad'e approximation problems along with their matrix generalizations. A general recurrence relation is given for solving these problems. Unlike other previous recursive methods, our recurrence works along arbitrary computational paths. When restricted to specific paths, the recurrence relation generalizes previous work of Antoulas, Cabay and Labahn, Beckermann, Van Barel and Bultheel and Gutknecht along with others.
On the Classification of the Spectrum of Second Order Difference Operators
, 1997
"... We study nonsymmetric second order difference operators acting in the Hilbert space ` 2 and describe the resolvent set and the essential spectrum of such operators in terms of related formal orthogonal polynomials. As an application, we obtain new results on the growth of orthonormal polynomials o ..."
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Cited by 4 (4 self)
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We study nonsymmetric second order difference operators acting in the Hilbert space ` 2 and describe the resolvent set and the essential spectrum of such operators in terms of related formal orthogonal polynomials. As an application, we obtain new results on the growth of orthonormal polynomials outside and inside the support of the underlying measure of orthogonality. 1.
On limit sets for the discrete spectrum of complex Jacobi matrices
 Mat. Sb
"... Abstract. The discrete spectrum of complex Jacobi matrices that are compact perturbations of the discrete laplacian is under consideration. The rate of stabilization for the matrix entries sharp in the sense of order which provides finiteness of the discrete spectrum is found. The Jacobi matrix with ..."
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Cited by 3 (2 self)
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Abstract. The discrete spectrum of complex Jacobi matrices that are compact perturbations of the discrete laplacian is under consideration. The rate of stabilization for the matrix entries sharp in the sense of order which provides finiteness of the discrete spectrum is found. The Jacobi matrix with the discrete spectrum having the only limit point is constructed. The results can be viewed as the discrete analogs of the well known theorems by B.S. Pavlov about Schrödinger operators on the half line with a complex potential.
Some Spectral Properties of Infinite Band Matrices.
, 2001
"... For operators generated by a certain class of ifinite band matrices we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order recurrence relations. This enables us to describe some asymptotic behaviour of the corresponding systems of vector ..."
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Cited by 2 (1 self)
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For operators generated by a certain class of ifinite band matrices we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order recurrence relations. This enables us to describe some asymptotic behaviour of the corresponding systems of vector orthogonal polynomials. Finally we provide some new convergence results for MatrixPadé approximants.
THE LINEAR PENCIL APPROACH TO RATIONAL INTERPOLATION
, 908
"... ABSTRACT. It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé approximants at infinity by considering rational interpolants, (bi)orthogonal rational functions and linear pencils zB − A of two tridiagonal matrices A, B, fol ..."
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ABSTRACT. It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé 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 Padé 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 ∞, we compute explicitly the spectrum and the numerical range of the underlying linear pencil. 1.