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, as well as 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 ~, we compute the spectrum and the numerical range of the underlying linear pencil explicitly.

[1]  B. Simon.  The Classical Moment Problem as a Self-Adjoint Finite Difference Operator , 1998, math-ph/9906008.

[2]  Adhemar Bultheel,et al.  Orthogonal Rational Functions , 1999, Cambridge monographs on applied and computational mathematics.

[3]  W. Assche,et al.  Criterion for the resolvent set of nonsymmetric tridiagonal operators , 1994, math/9401208.

[4]  A. Ambroladze.  On exceptional sets of asymptotic relations for general orthogonal polynomials , 1995 .

[5]  Mourad E. H. Ismail,et al.  Generalized orthogonality and continued fractions , 1994 .

[6]  B. Beckermann,et al.  The Diagonal of the Padé Table and the Approximation of the Weyl Function of Second-Order Difference Operators , 1997 .

[7]  A. Markus,et al.  Introduction to the Spectral Theory of Polynomial Operator Pencils , 2012 .

[8]  Francisco Marcellán,et al.  Darboux transformation and perturbation of linear functionals , 2004 .

[9]  Bernhard Beckermann,et al.  Complex Jacobi matrices , 2001 .

[10]  Vilmos Totik,et al.  General Orthogonal Polynomials , 1992 .

[11]  Maxim S. Derevyagin.  Generalized Jacobi operators in Krein spaces , 2008, 0806.1283.

[12]  Guillermo López Lagomasino,et al.  Asymptotic behavior of solutions of general three term recurrence relations , 2007, Adv. Comput. Math..

[13]  On the convergence of bounded J-fractions on the resolvent set of the corresponding second order dif , 1999 .

[14]  Alexei S. Zhedanov,et al.  TO THE THEORY OF BIORTHOGONAL RATIONAL FUNCTIONS (New Developments in the Research of Integrable Systems : Continuous, Discrete, Ultra-discrete) , 2003 .

[15]  L. Baratchart,et al.  Multipoint Schur algorithm and orthogonal rational functions, I: Convergence properties , 2008, 0812.2050.

[16]  Haakon Waadeland,et al.  Continued fractions with applications , 1994 .

[17]  William F. Moss,et al.  Decay rates for inverses of band matrices , 1984 .

[18]  Alexei Zhedanov.  Regular Article: Biorthogonal Rational Functions and the Generalized Eigenvalue Problem , 1999 .

[19]  Barry Simon,et al.  Orthogonal Polynomials on the Unit Circle , 2004, Encyclopedia of Special Functions: The Askey-Bateman Project.

[20]  V. N. Sorokin,et al.  Rational Approximations and Orthogonality , 1991 .

[21]  E. C. Titchmarsh,et al.  The theory of functions , 1933 .

[22]  Alexei S. Zhedanov,et al.  To the theory of biorthogonal rational functions , 2003 .

[23]  G. A. Baker,et al.  Pade approximants. Part 2: Extensions and applications , 1981 .

[24]  Maxim S. Derevyagin,et al.  On the convergence of Padé approximations for generalized Nevanlinna functions , 2007 .

[25]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[26]  Barry Simon,et al.  Orthogonal polynomials on the unit circle. Part 1 , 2005 .

[27]  On the determinacy of complex Jacobi matrices , 2004 .

[28]  Bernhard Beckermann,et al.  Some Spectral Properties of Infinite Band Matrices , 2003, Numerical Algorithms.

[29]  W. J. Thron,et al.  Continued Fractions: Analytic Theory and Applications , 1984 .

[30]  O. Njåstad,et al.  A favard theorem for rational functions , 1989 .

[31]  G. Szegő.  Zeros of orthogonal polynomials , 1939 .

[32]  B. Beckermann.  On the Classification of the Spectrum of Second Order Difference Operators , 2000 .

[33]  H. Wall,et al.  Analytic Theory of Continued Fractions , 2000 .

[34]  N. Akhiezer,et al.  The Classical Moment Problem. , 1968 .

[35]  Maxim S. Derevyagin,et al.  An operator approach to multipoint Padé approximations , 2008, J. Approx. Theory.