Abstract
A monic generalized Jacobi matrix \( \mathfrak{J} \) is factorized into upper and lower triangular two-diagonal block matrices of special forms so that J = UL. It is shown that such factorization depends on a free real parameter d(∈ ℝ). As the main result, it is shown that the matrix \( {\mathfrak{J}}^{\left(\mathbf{d}\right)}= LU \) is also a monic generalized Jacobi matrix. The matrix \( {\mathfrak{J}}^{\left(\mathbf{d}\right)} \) is called the Darboux transform of \( \mathfrak{J} \) with parameter d. An analog of the Geronimus formula for polynomials of the first kind of the matrix \( {\mathfrak{J}}^{\left(\mathbf{d}\right)} \) is proved, and the relations between m-functions of J and \( {\mathfrak{J}}^{\left(\mathbf{d}\right)} \) are found.
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References
N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver and Boyd, Edinburgh, 1965.
T. Ya. Azizov and I. S. Iokhvidov, Foundations of the Theory of Linear Operators in Spaces with an Indefinite Metric [in Russian], Nauka, Moscow, 1986.
Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators, Amer. Math. Soc., Providence, RI, 1969.
M. I. Bueno and F. Marcellán, “Darboux transformation and perturbation of linear functionals,” Lin. Algebra Appl., 384, 215–242 (2004).
M. Derevyagin and V. Derkach, “On convergence of Padé approximants for generalized Nevanlinna functions,” Trans. Moscow Math. Soc., 68, 133–182 (2007).
M. Derevyagin and V. Derkach, “Darboux transformations of Jacobi matrices and Padé approximation,” Lin. Algebra Appl., 435, 3056–3084 (2011).
M. Derevyagin and V. Derkach, “Spectral problems for generalized Jacobi matrices,” Lin. Algebra Appl., 382, 1–24 (2004).
V. A. Derkach and M. M. Malamud, “The extension theory of Hermitian operators and the moment problem,” J. Math. Sci., 73, 141–242 (1995).
V. A. Derkach and I. Kovalyov, “On a class of generalized Stieltjes continued fractions,” Methods Funct. Anal. Topol., 21, 315–335 (2015).
Ya. L. Geronimus, “On the polynomials orthogonal with respect to a given number sequence,” Zap. Mat. Otdel. Khar’kov. Univ., 17, 3–18 (1940).
Ya. L. Geronimus, “On the polynomials orthogonal with respect to a given number sequence and a theorem by W. Hahn,” Izv. Akad. Nauk SSSR, 4, 215–228 (1940).
F. Gesztesy and B. Simon, “m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices,” J. d’Analyse Math., 73, 267–297 (1997).
I. Kovalyov, “The Darboux transformation of generalized Jacobi matrices,” Methods Funct. Anal. Topol., 20, 301–320 (2014).
P. Lancaster, Theory of Matrices. Acad. Press, New York, 1969.
A. Magnus, “Certain continued fractions associated with the Padé table,” Math. Zeitschr., 78, 361–374 (1962).
F. Peherstorfer, “Finite perturbations of orthogonal polynomials,” J. Comput. Appl. Math., 44, 275–302 (1992).
G. Szegö, Orthogonal Polynomials, AMS, Providence, RI, 1975.
A. Zhedanov, “Rational spectral transformations and orthogonal polynomials,” J. Comput. Appl. Math., 85, 67–86 (1997).
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 13, No. 3, pp. 298–323 July–September, 2016.
This work was supported by a Volkswagen Stiftung grant and grants of the Ministry of Education and Science of Ukraine (projects 0115U000136 and 0115U000556)
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Kovalyov, I.M. Darboux transformation with parameter of generalized Jacobi matrices. J Math Sci 222, 703–722 (2017). https://doi.org/10.1007/s10958-017-3326-3
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DOI: https://doi.org/10.1007/s10958-017-3326-3