On the convergence of Padé approximations for generalized Nevanlinna functions

Derevyagin, M.; Derkach, V.

doi:10.1090/S0077-1554-07-00163-X

On the convergence of Padé approximations for generalized Nevanlinna functions
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by M. S. Derevyagin and V. A. Derkach
Translated by: O. A. Khleborodova

Trans. Moscow Math. Soc. 2007, 119-162

DOI: https://doi.org/10.1090/S0077-1554-07-00163-X

Published electronically: October 29, 2007

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Abstract:

We study a stepwise algorithm for solving the indefinite truncated moment problem and obtain the factorization of the matrix describing the solution of this problem into elementary factors. We consider the generalized Jacobi matrix corresponding to Magnus’ continuous $P$-fraction that appears in this algorithm and the polynomials of the first and second kind that are solutions of the corresponding difference equation. Weyl functions and the resolution matrices for finite and infinite Jacobi matrices are computed in terms of these polynomials. Convergence of diagonal and paradiagonal Padé approximation for functions from the generalized Nevanlinna class is studied.

References

A. I. Aptekarev, V. Kaliaguine, and W. Van Assche, Criterion for the resolvent set of nonsymmetric tridiagonal operators, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2423–2430. MR 1254830, DOI 10.1090/S0002-9939-1995-1254830-5
N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
D. Alpay, A. Dijksma, and H. Langer, Factorization of $J$-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions, Linear Algebra Appl. 387 (2004), 313–342. MR 2069282, DOI 10.1016/j.laa.2004.02.037
Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer, The Schur algorithm for generalized Schur functions. III. $J$-unitary matrix polynomials on the circle, Linear Algebra Appl. 369 (2003), 113–144. MR 1988481, DOI 10.1016/S0024-3795(02)00734-6
F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
T. Ya. Azizov and I. S. Iokhvidov, Linear operators in spaces with an indefinite metric, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1989. Translated from the Russian by E. R. Dawson; A Wiley-Interscience Publication. MR 1033489
Bernhard Beckermann, Complex Jacobi matrices, J. Comput. Appl. Math. 127 (2001), no. 1-2, 17–65. Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials. MR 1808568, DOI 10.1016/S0377-0427(00)00492-1
George A. Baker Jr., J. L. Gammel, and John G. Wills, An investigation of the applicability of the Padé approximant method, J. Math. Anal. Appl. 2 (1961), 405–418. MR 130093, DOI 10.1016/0022-247X(61)90019-1
George A. Baker Jr. and Peter Graves-Morris, Padé approximants. Part I, Encyclopedia of Mathematics and its Applications, vol. 13, Addison-Wesley Publishing Co., Reading, Mass., 1981. Basic theory; With a foreword by Peter A. Carruthers. MR 635619
Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718
V. I. Buslaev, The Baker-Gammel-Wills conjecture in the theory of Padé approximants, Mat. Sb. 193 (2002), no. 6, 25–38 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 5-6, 811–823. MR 1957951, DOI 10.1070/SM2002v193n06ABEH000658
F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Chelsea Publishing Co., New York, 1959. Translated by K. A. Hirsch. MR 0107649
Fritz Gesztesy and Barry Simon, $m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math. 73 (1997), 267–297. MR 1616422, DOI 10.1007/BF02788147
A. A. Gončar, The convergence of Padé approximants for certain classes of meromorphic functions, Mat. Sb. (N.S.) 97(139) (1975), no. 4(8), 607–629, 634 (Russian). MR 0387552
I. Gohberg, P. Lancaster, and L. Rodman, Matrix polynomials, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 662418
I. C. Gohberg and M. G. Kreĭn, Fundamental aspects of defect numbers, root numbers and indexes of linear operators, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 2(74), 43–118 (Russian). MR 0096978
V. I. Gorbachuk and M. L. Gorbachuk, Boundary value problems for operator differential equations, Mathematics and its Applications (Soviet Series), vol. 48, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated and revised from the 1984 Russian original. MR 1154792, DOI 10.1007/978-94-011-3714-0
A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, A factorization result for generalized Nevanlinna functions of the class $\scr N_\kappa$, Integral Equations Operator Theory 36 (2000), no. 1, 121–125. MR 1736921, DOI 10.1007/BF01236290
M. Derevyagin, On the Schur algorithm for indefinite moment problem, Methods Funct. Anal. Topology 9 (2003), no. 2, 133–145. MR 1999775
Maxim Derevyagin and Vladimir Derkach, Spectral problems for generalized Jacobi matrices, Linear Algebra Appl. 382 (2004), 1–24. MR 2050096, DOI 10.1016/j.laa.2003.11.003
V. A. Derkach, On generalized resolvents of Hermitian relations in Krein spaces, J. Math. Sci. (New York) 97 (1999), no. 5, 4420–4460. Functional analysis, 5. MR 1728871, DOI 10.1007/BF02366102
V. A. Derkach, On characteristic functions of linear relations and unitary colligations, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 11 (2001), 28–33 (English, with Ukrainian summary). MR 1900470
Vladimir Derkach, Seppo Hassi, and Henk de Snoo, Operator models associated with Kac subclasses of generalized Nevanlinna functions, Methods Funct. Anal. Topology 5 (1999), no. 1, 65–87. MR 1771251
Vladimir Derkach, Seppo Hassi, and Henk De Snoo, Generalized Nevanlinna functions with polynomial asymptotic behaviour at infinity and regular perturbations, Operator theory and analysis (Amsterdam, 1997) Oper. Theory Adv. Appl., vol. 122, Birkhäuser, Basel, 2001, pp. 169–189. MR 1846057
V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (1995), no. 2, 141–242. Analysis. 3. MR 1318517, DOI 10.1007/BF02367240
William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864
Harry Dym, On Hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy, Integral Equations Operator Theory 12 (1989), no. 6, 757–812. MR 1018213, DOI 10.1007/BF01196878
I. S. Iohvidov, M. G. Kreĭn, and H. Langer, Introduction to the spectral theory of operators in spaces with an indefinite metric, Mathematical Research, vol. 9, Akademie-Verlag, Berlin, 1982. MR 691137
M. G. Kreĭn, The fundamental propositions of the theory of representations of Hermitian operators with deficiency index $(m,m)$, Ukrain. Mat. Žurnal 1 (1949), no. 2, 3–66 (Russian). MR 0048704
M. G. Kreĭn and G. K. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space $\Pi _{\kappa }$, Funkcional. Anal. i Priložen. 5 (1971), no. 2, 59–71 (Russian). MR 0282238
M. G. Kreĭn and H. Langer, Über die $Q$-Funktion eines $\pi$-hermiteschen Operators im Raume $\Pi _{\kappa }$, Acta Sci. Math. (Szeged) 34 (1973), 191–230 (German). MR 318958
M. G. Kreĭn and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume $\Pi _{\kappa }$ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236. MR 461188, DOI 10.1002/mana.19770770116
M. G. Kreĭn and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume $\Pi _{\kappa }$ zusammenhängen. II. Verallgemeinerte Resolventen, $u$-Resolventen und ganze Operatoren, J. Functional Analysis 30 (1978), no. 3, 390–447 (German). MR 518342, DOI 10.1016/0022-1236(78)90064-2
M. G. Kreĭn and Heinz Langer, On some extension problems which are closely connected with the theory of Hermitian operators in a space $\Pi _{\kappa }$. III. Indefinite analogues of the Hamburger and Stieltjes moment problems. Part I, Beiträge Anal. 14 (1979), 25–40 (loose errata). MR 563344
M. G. Kreĭn and H. Langer, Some propositions on analytic matrix functions related to the theory of operators in the space $\Pi _{\kappa }$, Acta Sci. Math. (Szeged) 43 (1981), no. 1-2, 181–205. MR 621369
M. G. Kreĭn and A. A. Nudel′man, The Markov moment problem and extremal problems, Translations of Mathematical Monographs, Vol. 50, American Mathematical Society, Providence, R.I., 1977. Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development; Translated from the Russian by D. Louvish. MR 0458081
M. G. Kreĭn and Š. N. Saakjan, The resolvent matrix of a Hermitian operator and the characteristic functions connected with it, Funkcional. Anal. i Priložen. 4 (1970), no. 3, 103–104 (Russian). MR 0275205
Heinz Langer, A characterization of generalized zeros of negative type of functions of the class $N_\kappa$, Advances in invariant subspaces and other results of operator theory (Timişoara and Herculane, 1984) Oper. Theory Adv. Appl., vol. 17, Birkhäuser, Basel, 1986, pp. 201–212. MR 901070, DOI 10.1007/978-3-0348-7698-8_{1}5
H. Langer and B. Textorius, On generalized resolvents and $Q$-functions of symmetric linear relations (subspaces) in Hilbert space, Pacific J. Math. 72 (1977), no. 1, 135–165. MR 463964
D. S. Lubinsky, Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture, Ann. of Math. (2) 157 (2003), no. 3, 847–889. MR 1983783, DOI 10.4007/annals.2003.157.847
Arne Magnus, Certain continued fractions associated with the Padé table, Math. Z. 78 (1962), 361–374. MR 150271, DOI 10.1007/BF01195180
Arne Magnus, Expansion of power series into $P$-fractions, Math. Z. 80 (1962), 209–216. MR 150272, DOI 10.1007/BF01162378
E. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, Translations of Mathematical Monographs, vol. 92, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Ralph P. Boas. MR 1130396, DOI 10.1090/mmono/092
E. A. Rahmanov, The convergence of diagonal Padé approximants, Mat. Sb. (N.S.) 104(146) (1977), no. 2(10), 271–291, 335 (Russian). MR 0492292
Barry Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82–203. MR 1627806, DOI 10.1006/aima.1998.1728
Herbert Stahl, On the divergence of certain Padé approximant and the behaviour of the associated orthogonal polynomials, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 321–330. MR 839001, DOI 10.1007/BFb0076561
H. van Rossum, Padé approximants and indefinite inner product spaces, Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New York, 1977, pp. 111–119. MR 0617938