Abstract
For operators generated by a certain class of infinite 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 matrix Padé approximants.
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References
A. Aptekarev, V. Kaliaguine and J. Van Iseghem, Genetic sums representation for the moments of a system of Stieltjes functions and its applications, Construct. Approx. 16 (2000) 487–524.
B. Beckermann, Nonsymmetric difference operators and polynomials being orthogonal with respect to rectangular matrix valued measures, Publication ANO 335, Université de Lille (1995).
B. Beckermann, On the convergence of bounded J-fractions on the resolvent set of the corresponding second order difference operator, J. Approx. Theory 99 (1999) 369–408.
B. Beckermann, On the classification of the spectrum of second order difference operators, Math. Nachrichten 216 (2000) 45–59.
B. Beckermann, Complex Jacobi matrices, J. Comput. Appl. Math. 127 (2001) 17–65.
B. Beckermann and M. Castro, On the determinacy of complex Jacobi matrices, Publication ANO 446, Université de Lille (2002) (submitted).
B. Beckermann and V. A. Kaliaguine, The diagonal of the Padé table and the approximation of the Weyl function of second order difference operators, Construct. Approx. 13 (1997) 481–510.
Yu.M. Berezanskii, Integration of nonlinear difference equation by the inverse spectral problem method, Dokl. Soviet Acad. Nauk 281 (1985). English transl. in Soviet Math. Dokl. 31 (1985) 264–267.
S. Demko, W.F. Moss and P.W. Smith, Decay rates for inverses of band matrices, Math. Comp. 43 (1984) 491–499.
V.A. Kaliaguine, Hermite–Padé approximants and spectral analysis of nonsymmetric operators, Mat. Sb. 185 (1994) 79–100 (in Russian). English transl. in Russian Acad. Sci. Sb. Math. 82 (1995) 199–216.
Ju.L. Kishakevich, Spectral function of Marchenko type for a difference operator of an even order, Math. Notes 11 (1972) 266–271.
A.G. Kostyuchenko and K.A. Mirzoev, Generalized Jacobi matrices and deficiency indices of ordinary differential operators with polynomial coefficients, Funct. Anal. Appl. 33 (1999) 25–37.
K. Mahler, Perfect systems, Compos. Math. 19 (1968) 95–166.
V. N. Sorokin and J. Van Iseghem, Algebraic aspects of matrix orthogonality for vector polynomials, J. Approx. Theory 90 (1997) 97–116.
V.N. Sorokin and J. Van Iseghem, Matrix continued fractions, J. Approx. Theory 96 (1998) 237–257.
V.N. Sorokin and J. Van Iseghem, Matrix Hermite–Padé problem and dynamical systems, J. Comput. Appl. Math. 122 (2000) 275–295.
V.A. Yurko, On higher-order difference operators, J. Difference Equations Appl. 1 (1996) 347–352.
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Beckermann, B., Osipov, A. Some Spectral Properties of Infinite Band Matrices. Numerical Algorithms 34, 173–185 (2003). https://doi.org/10.1023/B:NUMA.0000005361.17723.a4
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DOI: https://doi.org/10.1023/B:NUMA.0000005361.17723.a4