Abstract
In the PhD thesis of the second author under the supervision of the third author was defined the class \({\mathcal{SI}}\) of J-contractive functions, depending on a parameter and arising as transfer functions of overdetermined conservative 2D systems invariant in one direction. In this paper we extend and solve in the class \({\mathcal{SI}}\), a number of problems originally set for the class \({\mathcal{S}}\) of functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned signature matrix J. The problems we consider include the Schur algorithm and the Nevanlinna–Pick interpolation problem. The arguments rely on a correspondence between elements in a given subclass of \({\mathcal{SI}}\) and elements in \({\mathcal{S}}\). Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.
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References
Alpay, D.: Schur Algorithm, Reproducing Kernel Hilbert Spaces and the System Theory. Translated from the 1998 French original by Stephen S. Wilson, Panoramas et Synthèses. American Mathematical Society, Providence (2001)
Alpay D., Dijksma A., Rovnyak J., de Snoo H.: Schur functions, operator colligations, and reproducing kernel Pontryagin spaces. In: Operator Theory: Advances and Applications, vol. 96. Birkhüser, Basel (1997)
Alpay, D., Dym, H.: On applications of reproducing kernel spaces to the Schur algorithm and rational j-unitary factorizations. In: Operator Theory: Advances and Applications, vol. 347, issue OT18, pp. 89–159. Birkhäuser, Basel (1986)
Alpay D., Dym H.: On a new class of structured reproducing kernel spaces. J. Funct. Anal. 111(1), 1–28 (1993)
Alpay, D., Dym, H.: On a new class of realization formulas and their application. In: Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994), vol. 241/243, pp. 3–84 (1996)
Alpay, D., Gohberg, I.: Unitary rational matrix functions. Topics in interpolation theory of rational matrix-valued functions. In: Operator Theory: Advances and Applications, vol. 33, pp. 175–222. Birkhüser, Basel (1988)
Alpay D., Melnikov A., Vinnikov V.: Un algorithme de Schur pour les fonctions de transfert des systèmes surdéterminés invariants dans une direction. Comptes-Rendus mathématiques (Paris) 347(13–14), 729–733 (2009)
Ball, J., Cohen N.: de Branges–Rovnyak operator models and system theory: a survey. In: Operator Theory: Advances and Applications, vol. 50, pp. 93–136. Birkhäuser, Basel (1991)
Ball J., Gohberg I., Rodman L.: Interpolation of rational matrix functions. Operator Theory: Advances and Applications. Birkhäuser, Basel (1990)
Bart H., Gohberg I., Kaashoek M.A.: Minimal factorization of matrix and operator functions, vol. 1. Operator Theory: Advances and Applications. Birkhäuser, Basel (1979)
de Branges, L., Rovnyak, J.: Canonical models in quantum scattering theory. In: Perturbation Theory and Its Applications in Quantum Mechanics, pp. 295–392 (1966)
de Branges L., Rovnyak J.: Square summable power series. Holt, Rinehart and Winston, New York (1966)
Coddington E.A., Levinson N.: Theory of ordinary differential eqnarrays. Mc-Graw Hill, New York (1955)
Dewilde P., Dym H.: Lossless inverse scattering, digital filters, and estimation theory. IEEE Trans. Inform. Theory 30, 644–662 (1984)
Donoghue W.F.: Monotone matrix functions and analytic continuation. Springer, Berlin (1974)
Dunford N., Schwartz J.T.: Linear Operators, General Theory. Wiley, New York (1988)
Dym, H.: J–contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, vol. 71. American Mathematical Society, Providence (1989). Conference Board of the Mathematical Sciences, Regional conference series in Mathematics (1989)
Fritzsche B., Kirstein B.: Ausgewählte Arbeiten zu den Ursprüngen der Schur–Analysis, vol. 16. Teubner-Archiv zur Mathematik, Reihe (1991)
Helton J.W.: Discrete time systems, operator models, and scattering theory. J. Funct. Anal. 16, 15–38 (1974)
Livšic M.S.: Operator colligations, waves, open systems. Transl. Math. Monog. AMS, Providence (1973)
Livšic, M.S.: Vortices of 2D systems. In: Operator Theory: Advances and Applications, Birkhäuser, Basel vol. 123, pp. 7–41. (2001)
Livšic M.S., Brodskii M.S.: Spectral analysis of non-self-adjoint operators and intermediate systems (Russian). Uspehi Mat. Nauk (N.S.) 13(1(79)), 3–85 (1958)
Melnikov, A.: Overdetermined 2D systems invariant in one direction and their transfer functions. PhD thesis, Ben Gurion University (2009)
Melnikov A.: Finite dimensional Sturm Liouville vessels and their tau functions. IEOT 74(4), 455–490 (2011)
Melnikov, A., Vinnikov, V.: Overdetermined 2D systems invariant in one direction and their transfer functions. http://arXiv.org/abs/0812.3779
Melnikov, A., Vinnikov, V.: Overdetermined conservative 2D systems, invariant in one direction and a generalization of Potapov’s theorem. http://arxiv.org/abs/0812.3970
Nevanlinna R.: Über beschränkte Funktionen, die in gegebenen Punkten vorgeschriebene Werte annehmen. Ann. Acad. Sci. Fenn. 1, 1–71 (1919)
Pick G.: Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden. Math. Ann. 77 1, 7–23 (1915)
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D. Alpay thanks the Earl Katz family for endowing the chair which supported his research and the Binational Science Foundation Grant number 2010117. The research of the authors was supported in part by the Israel Science Foundation grant 1023/07.
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Alpay, D., Melnikov, A. & Vinnikov, V. Schur Algorithm in The Class \({\mathcal{SI}}\) of J-contractive Functions Intertwining Solutions of Linear Differential Equations. Integr. Equ. Oper. Theory 74, 313–344 (2012). https://doi.org/10.1007/s00020-012-2002-8
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DOI: https://doi.org/10.1007/s00020-012-2002-8