Book Review
The AMS does not provide abstracts of book reviews.
You may download the entire review from the links below.
MathSciNet review:
1568113
Full text of review:
PDF
This review is available free of charge.
Book Information:
Author:
Ciprian Foias and Arthur E. Frazho
Title:
The commutant lifting approach to interpolation problems
Additional book information:
Birkh\"auser, Basel, 1990, 6331 pp., US$129{.}00. ISBN 3-7643-2461-9.
V. M. Adamjan, Nondegenerate unitary couplings of semiunitary operators, Funkcional. Anal. i Priložen. 7 (1973), no. 4, 1–16 (Russian). MR 0343066
V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Infinite Hankel matrices and generalized Carathéodory-Fejér and I. Schur problems, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 1–17 (Russian). MR 0636333
[AAK2] -, Analytic properties of Schmidt pairs for a Hankel operator and the generalized problem of Schur-Takagi, USSR-Sb. 15 (1971), 31-73.
V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Infinite Hankel block matrices and related problems of extension, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 2-3, 87–112 (Russian, with Armenian and English summaries). MR 0298454
[Ak] N. I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh, 1965.
D. Alpay and V. Bolotnikov, Two-sided Nevanlinna-Pick interpolation for a class of matrix-valued functions, Z. Anal. Anwendungen 12 (1993), no. 2, 211–238. MR 1245916, DOI 10.4171/ZAA/570
Daniel Alpay, Piet Bruinsma, Aad Dijksma, and Henk de Snoo, Interpolation problems, extensions of symmetric operators and reproducing kernel spaces. I, Topics in matrix and operator theory (Rotterdam, 1989) Oper. Theory Adv. Appl., vol. 50, Birkhäuser, Basel, 1991, pp. 35–82. MR 1115024, DOI 10.1007/978-3-0348-5672-0_{3}
Daniel Alpay, Piet Bruinsma, Aad Dijksma, and Henk de Snoo, Interpolation problems, extensions of symmetric operators and reproducing kernel spaces. I, Topics in matrix and operator theory (Rotterdam, 1989) Oper. Theory Adv. Appl., vol. 50, Birkhäuser, Basel, 1991, pp. 35–82. MR 1115024, DOI 10.1007/978-3-0348-5672-0_{3}
Daniel Alpay and Harry Dym, On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains, Operator theory and complex analysis (Sapporo, 1991) Oper. Theory Adv. Appl., vol. 59, Birkhäuser, Basel, 1992, pp. 30–77. MR 1246809
Daniel Alpay and Harry Dym, On a new class of structured reproducing kernel spaces, J. Funct. Anal. 111 (1993), no. 1, 1–28. MR 1200633, DOI 10.1006/jfan.1993.1001
D. Z. Arov, Gamma-generating matrices, $j$-inner matrix functions and related extrapolation problems. I, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 51 (1989), 61–67 (Russian); English transl., J. Soviet Math. 52 (1990), no. 6, 3487–3491. MR 1009146, DOI 10.1007/BF01095407
D. Z. Arov, Regular $J$-inner matrix-functions and related continuation problems, Linear operators in function spaces (Timişoara, 1988) Oper. Theory Adv. Appl., vol. 43, Birkhäuser, Basel, 1990, pp. 63–87. MR 1090117
D. Z. Arov and L. Z. Grossman, Scattering matrices in the theory of unitary extension of isometric operators, Math. Nachr. 157 (1992), 105–123 (English, with German summary). MR 1233051, DOI 10.1002/mana.19921570110
D. Z. Arov and M. G. Kreĭn, Calculation of entropy functionals and their minima in indeterminate continuation problems, Acta Sci. Math. (Szeged) 45 (1983), no. 1-4, 33–50 (Russian). MR 708770
Rodrigo Arocena, Generalized Toeplitz kernels and dilations of intertwining operators, Integral Equations Operator Theory 6 (1983), no. 6, 759–778. MR 719104, DOI 10.1007/BF01691923
Rodrigo Arocena, Generalized Toeplitz kernels and dilations of intertwining operators. II. The continuous case, Acta Sci. Math. (Szeged) 53 (1989), no. 1-2, 123–137. MR 1018681
[BaC] M. Bakonyi and T. Constantinescu, Schur's algorithm and several applications, Pitman Res. Notes Math. Ser., vol. 61, Longman, Essex, 1992.
Joseph A. Ball, Models for noncontractions, J. Math. Anal. Appl. 52 (1975), no. 2, 235–254. MR 388135, DOI 10.1016/0022-247X(75)90093-1
Joseph A. Ball, Nevanlinna-Pick interpolation: generalizations and applications, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 51–94. MR 958570
Joseph A. Ball and Nir Cohen, Sensitivity minimization in an $H^\infty$ norm: parametrization of all suboptimal solutions, Internat. J. Control 46 (1987), no. 3, 785–816. MR 908751, DOI 10.1080/00207178708547393
Joseph A. Ball, Israel Gohberg, and Leiba Rodman, Interpolation of rational matrix functions, Operator Theory: Advances and Applications, vol. 45, Birkhäuser Verlag, Basel, 1990. MR 1083145, DOI 10.1007/978-3-0348-7709-1
Joseph A. Ball and J. William Helton, Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parametrization of the set of all solutions, Integral Equations Operator Theory 9 (1986), no. 2, 155–203. MR 837514, DOI 10.1007/BF01195006
Joseph A. Ball and André C. M. Ran, Optimal Hankel norm model reductions and Wiener-Hopf factorization. II. The noncanonical case, Integral Equations Operator Theory 10 (1987), no. 3, 416–436. MR 883496, DOI 10.1007/BF01195036
Louis de Branges, Some Hilbert spaces of analytic functions. I, Trans. Amer. Math. Soc. 106 (1963), 445–468. MR 145335, DOI 10.1090/S0002-9947-1963-0145335-7
Piet Bruinsma, Degenerate interpolation problems for Nevanlinna pairs, Indag. Math. (N.S.) 2 (1991), no. 2, 179–200. MR 1123360, DOI 10.1016/0019-3577(91)90005-R
[BD] A. Bultheel and P. Dewilde, On the Adamjan-Arov-Krein approximation, identification and balanced realization of a system, Technical Report TW48, Katholieke Universiteit Leuven, 1980; a short version appears in Proceedings ECTI (1980), 186-191.
Bor Chin Chang and J. Boyd Pearson Jr., Optimal disturbance reduction in linear multivariable systems, IEEE Trans. Automat. Control 29 (1984), no. 10, 880–887. MR 760526, DOI 10.1109/TAC.1984.1103409
Mischa Cotlar and Cora Sadosky, Toeplitz liftings of Hankel forms, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 22–43. MR 942255, DOI 10.1007/BFb0078861
[CZ] R. F. Curtain and H. J. Zwart, The Nehari problem or the Pritchard-Salmon class of infinite-dimensional linear system: A direct approach, preprint, 1993.
Ph. Delsarte, Y. Genin, and Y. Kamp, On the role of the Nevanlinna-Pick problem in circuit and system theory, Internat. J. Circuit Theory Appl. 9 (1981), no. 2, 177–187. MR 612269, DOI 10.1002/cta.4490090204
Patrick Dewilde and Harry Dym, Lossless chain scattering matrices and optimum linear prediction: the vector case, Internat. J. Circuit Theory Appl. 9 (1981), no. 2, 135–175. MR 612268, DOI 10.1002/cta.4490090203
John C. Doyle, Keith Glover, Pramod P. Khargonekar, and Bruce A. Francis, State-space solutions to standard ${\scr H}_2$ and ${\scr H}_\infty$ control problems, IEEE Trans. Automat. Control 34 (1989), no. 8, 831–847. MR 1004301, DOI 10.1109/9.29425
Harry Dym, $J$ contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS Regional Conference Series in Mathematics, vol. 71, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. MR 1004239, DOI 10.1090/cbms/071
Harry Dym, On reproducing kernel spaces, $J$ unitary matrix functions, interpolation and displacement rank, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988) Oper. Theory Adv. Appl., vol. 41, Birkhäuser, Basel, 1989, pp. 173–239. MR 1038337
Harry Dym, On reproducing kernels and the continuous covariance extension problem, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 427–482. MR 1044802
[D4] -, Shifts, realizations and interpolation, redux, Operator Theory and its Applications (A. Feintuch and I. Gohberg, eds.), Oper. Theory Adv. Appl., Birkhäuser-Verlag, Basel, 1994.
Harry Dym, On the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares, Comm. Pure Appl. Math. 47 (1994), no. 2, 207–256. MR 1263129, DOI 10.1002/cpa.3160470205
Vladimir K. Dubovoj, Bernd Fritzsche, and Bernd Kirstein, Matricial version of the classical Schur problem, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 129, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992. With German, French and Russian summaries. MR 1152328
R. L. Ellis, I. Gohberg, and D. C. Lay, Extensions with positive real part. A new version of the abstract band method with applications, Integral Equations Operator Theory 16 (1993), no. 3, 360–384. MR 1209305, DOI 10.1007/BF01204227
[FFHKP] C. Foias, B. Francis, J. W. Helton, H. Kwakernaak, and J. B. Pearson, -control theory (E. Mosca and L. Pandolfi, eds.), Lecture Notes in Math., vol. 1496, Springer-Verlag, Berlin, 1991.
Ciprian Foias and Arthur E. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, vol. 44, Birkhäuser Verlag, Basel, 1990. MR 1120546, DOI 10.1007/978-3-0348-7712-1
Ciprian Foias and Allen Tannenbaum, On the four-block problem. I, Topics in operator theory, Oper. Theory Adv. Appl., vol. 32, Birkhäuser, Basel, 1988, pp. 93–112. MR 951958, DOI 10.1007/978-3-0348-5475-7_{8}
[Fr] B. A. Francis, A course in control theory, Lecture Notes in Control and Inform. Sci., vol. 88, Springer-Verlag, Berlin, 1987.
Bruce A. Francis and John C. Doyle, Linear control theory with an $H_\infty$ optimality criterion, SIAM J. Control Optim. 25 (1987), no. 4, 815–844. MR 893986, DOI 10.1137/0325046
Bruce A. Francis and George Zames, On $H^\infty$-optimal sensitivity theory for SISO feedback systems, IEEE Trans. Automat. Control 29 (1984), no. 1, 9–16. MR 734241, DOI 10.1109/TAC.1984.1103357
Yves V. Genin and Sun Yuan Kung, A two-variable approach to the model reduction problem with Hankel norm criterion, IEEE Trans. Circuits and Systems 28 (1981), no. 9, 912–924. MR 629917, DOI 10.1109/TCS.1981.1085067
Tryphon T. Georgiou and Pramod P. Khargonekar, Spectral factorization and Nevanlinna-Pick interpolation, SIAM J. Control Optim. 25 (1987), no. 3, 754–766. MR 885197, DOI 10.1137/0325043
Tryphon T. Georgiou and Malcolm C. Smith, Topological approaches to robustness, Analysis and optimization of systems: state and frequency domain approaches for infinite-dimensional systems (Sophia-Antipolis, 1992) Lect. Notes Control Inf. Sci., vol. 185, Springer, Berlin, 1993, pp. 222–241. MR 1208272, DOI 10.1007/BFb0115027
[Glo] K. Glover, A tutorial on Hankel norm approximation, From Data to Model (J. C. Willems, ed.), Springer, Berlin, 1989, pp. 26-48.
K. Glover and J. C. Doyle, A state space approach to $H_\infty$ optimal control, Three decades of mathematical system theory, Lect. Notes Control Inf. Sci., vol. 135, Springer, Berlin, 1989, pp. 179–218. MR 1025791, DOI 10.1007/BFb0008463
K. Glover, D. J. N. Limebeer, J. C. Doyle, E. M. Kasenally, and M. G. Safonov, A characterization of all solutions to the four block general distance problem, SIAM J. Control Optim. 29 (1991), no. 2, 283–324. MR 1092729, DOI 10.1137/0329016
Michael Green, ${\scr H}_\infty$ controller synthesis by $J$-lossless coprime factorization, SIAM J. Control Optim. 30 (1992), no. 3, 522–547. MR 1160141, DOI 10.1137/0330031
Michael Green, Keith Glover, David Limebeer, and John Doyle, A $J$-spectral factorization approach to $\scr H_\infty$ control, SIAM J. Control Optim. 28 (1990), no. 6, 1350–1371. MR 1075207, DOI 10.1137/0328071
Michael Green and David J. N. Limebeer, Vector interpolation, $\scr H_\infty$ control and model reduction, Robust control of linear systems and nonlinear control (Amsterdam, 1989) Progr. Systems Control Theory, vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 285–292. MR 1115399
J. William Helton, Orbit structure of the Möbius transformation semigroup acting on $H^{\infty }$ (broadband matching), Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), Adv. in Math. Suppl. Stud., vol. 3, Academic Press, New York-London, 1978, pp. 129–157. MR 538017
J. William Helton, Joseph A. Ball, Charles R. Johnson, and John N. Palmer, Operator theory, analytic functions, matrices, and electrical engineering, CBMS Regional Conference Series in Mathematics, vol. 68, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1987. MR 896034, DOI 10.1090/cbms/068
G. Herglotz, I. Schur, G. Pick, R. Nevanlinna, and H. Weyl, Ausgewählte Arbeiten zu den Ursprüngen der Schur-Analysis, Teubner-Archiv zur Mathematik [Teubner Archive on Mathematics], vol. 16, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991 (German). Gewidmet dem großen Mathematiker Issai Schur (1875–1941). [Dedicated to the great mathematician Issai Schur (1875–1941)]; Edited and with a foreword and afterword by Bernd Fritzsche and Bernd Kirstein; With contributions by W. Ledermann and W. K. Hayman; With English, French and Russian summaries. MR 1162976, DOI 10.1007/978-3-322-90926-8
V. È. Katsnel′son, Methods of $J$-theory in continuous interpolation problems of analysis. Part I, T. Ando, Hokkaido University, Sapporo, 1985. Translated from the Russian and with a foreword by T. Ando. MR 777324
V. È. Katsnel′son, The problem of completion of a partial matrix function as a classical interpolation problem, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 56 (1991), 105–122 (Russian); English transl., J. Math. Sci. 76 (1995), no. 4, 2525–2537. MR 1220900, DOI 10.1007/BF02364910
V. È. Katsnel′son, A. Ya. Kheĭfets, and P. M. Yuditskiĭ, An abstract interpolation problem and the theory of extensions of isometric operators, Operators in function spaces and problems in function theory (Russian), “Naukova Dumka”, Kiev, 1987, pp. 83–96, 146 (Russian). MR 946502
Hidenori Kimura, Directional interpolation approach to $H^\infty$-optimization and robust stabilization, IEEE Trans. Automat. Control 32 (1987), no. 12, 1085–1093. MR 914841, DOI 10.1109/TAC.1987.1104504
Hidenori Kimura, Conjugation, interpolation and model-matching in $H^\infty$, Internat. J. Control 49 (1989), no. 1, 269–307. MR 981729, DOI 10.1080/00207178908961244
H. Kimura, State space approach to the classical interpolation problem and its applications, Three decades of mathematical system theory, Lect. Notes Control Inf. Sci., vol. 135, Springer, Berlin, 1989, pp. 243–275. MR 1025793, DOI 10.1007/BFb0008465
Sun Yuan Kung and David W. Lin, Optimal Hankel-norm model reductions: multivariable systems, IEEE Trans. Automat. Control 26 (1981), no. 4, 832–852. MR 635843, DOI 10.1109/TAC.1981.1102736
I. V. Kovalishina, Analytic theory of a class of interpolation problems, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 3, 455–497 (Russian). MR 703593
[KL1] M. G. Krein and H. Langer, On some extension problems which are closely connected with the theory of hermitian operators in a space , III. Indefinite analogues of the Hamburger and Stieltjes moment problems, Beiträge zur Analysis 14 (1979), 25-40; 15 (1980), 27-45.
Mark G. Kreĭn and Heinz Langer, On some continuation problems which are closely related to the theory of operators in spaces $\Pi _\kappa$. IV. Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions, J. Operator Theory 13 (1985), no. 2, 299–417. MR 776000
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
D. J. N. Limebeer and B. D. O. Anderson, An interpolation theory approach to $H^\infty$ controller degree bounds, Linear Algebra Appl. 98 (1988), 347–386. MR 919123, DOI 10.1016/0024-3795(88)90172-3
David J. N. Limebeer and Michael Green, Parametric interpolation, $H^\infty$-control and model reduction, Internat. J. Control 52 (1990), no. 2, 293–318. MR 1061723, DOI 10.1080/00207179008953538
[LK] D. J. N. Limebeer and E. M. Kasenelly, Necessary and sufficient conditions for the existence of -optimal controllers; an interpolation approach, Robust Control of Linear Systems and Nonlinear Control (M. A. Kaashoek, J. H. van Schuppen, and A. C. M. Ran, eds.), Birkhäuser, Boston, 1990, pp. 309-316.
[Nu] A. A. Nudelman, On a new problem of moment type, Soviet Math. Doklady 18 (1977), 507-510.
Lavon B. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc. 150 (1970), 529–539. MR 273449, DOI 10.1090/S0002-9947-1970-0273449-3
Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. Oxford Science Publications. MR 822228
Donald Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179–203. MR 208383, DOI 10.1090/S0002-9947-1967-0208383-8
Béla Sz.-Nagy and Ciprian Foiaş, Commutants de certains opérateurs, Acta Sci. Math. (Szeged) 29 (1968), 1–17 (French). MR 242011
[SNF2] -, Dilation des commutants d'opérateurs, C. R. Acad. Sci. Paris Sér. I Math. 266 (1968), 493-495.
[SNF3] -, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970.
Allen Tannenbaum, Modified Nevalinna-Pick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor, Internat. J. Control 36 (1982), no. 2, 331–336. MR 674395, DOI 10.1080/00207178208932897
Gilead Tadmor, An interpolation problem associated with $H^\infty$-optimal design in systems with distributed input lags, Systems Control Lett. 8 (1987), no. 4, 313–319. MR 884880, DOI 10.1016/0167-6911(87)90097-1
Gilead Tadmor, The standard $H_\infty$ problem and the maximum principle: the general linear case, SIAM J. Control Optim. 31 (1993), no. 4, 813–846. MR 1227535, DOI 10.1137/0331036
D. C. Youla and M. Saito, Interpolation with positive-real functions, J. Franklin Inst. 284 (1967), 77–108. MR 255824, DOI 10.1016/0016-0032(67)90582-0
N. J. Young, The Nevanlinna-Pick problem for matrix-valued functions, J. Operator Theory 15 (1986), no. 2, 239–265. MR 833210
G. Zames, Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses, IEEE Trans. Automat. Control 26 (1981), no. 2, 301–320. MR 613539, DOI 10.1109/TAC.1981.1102603
- [Ad]
- V. M. Adamyan, Nondegenerate unitary couplings of semiunitary operators, Funktsional. Anal. i Prilozhen 7 (1973), 1-16; English transl., Functional Anal. Appl. 7 (1973), 255-267. MR 0343066 (49:7810)
- [AAK1]
- V. M. Adamjan, D. Z. Arov, and M. G. Krein, Infinite Hankel matrices and generalized Carathéodory-Fejér and I. Schur problems, Funktsional. Anal. i Prilozhen 2 (1968), 1-17. MR 0636333 (58:30446)
- [AAK2]
- -, Analytic properties of Schmidt pairs for a Hankel operator and the generalized problem of Schur-Takagi, USSR-Sb. 15 (1971), 31-73.
- [AAK3]
- -, Infinite block Hankel matrices and problems of continuation related to them, Akad. Nauk Armenia SSR Izv. Mat. 6 (1971), 87-112; English transl., Amer. Math. Soc. Transl. Ser. 2 111 (1978), 133-156. MR 0298454 (45:7506)
- [Ak]
- N. I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh, 1965.
- [AB]
- D. Alpay and V. Bolotnikov, Two sided Nevanlinna-Pick interpolation for a class of matrix valued functions, Z. Anal. Anwendungen 12 (1993), 221-238. MR 1245916 (95d:47010)
- [ABDS1]
- D. Alpay, P. Bruinsma, A. Dijksma, and H. de Snoo, Interpolation problems, extensions of symmetric operators and reproducing kernel spaces I, Topics in Matrix and Operator Theory (H. Bart, I. Gohberg, and M. Kaashoek, eds.), Oper. Theory Adv. Appl., vol. 50, Birkhäuser, Basel, 1991, pp. 35-82. MR 1115024 (93k:47026a)
- [ABDS2]
- -, Interpolation problems, extensions of symmetric operators and reproducing kernel spaces. II, Integral Equations Operator Theory 14 (1991), 465-500; 15 (1992), 378-388. MR 1109834 (93k:47026b)
- [AD1]
- D. Alpay and H. Dym, On reproducing kernels spaces, the Schur algorithm and interpolation in a general class of domains, Operator Theory and Complex Analysis (T. Ando and I. Gohberg, eds.), Oper. Theory Adv. Appl., vol. 59, Birkhäuser-Verlag, Basel, 1992, pp. 30-77. MR 1246809 (94j:46034)
- [AD2]
- -, On a new class of structured reproducing kernel spaces, J. Funct. Anal. 111 (1993), 1-28. MR 1200633 (94g:46035)
- [Arov1]
- D. Z. Arov, Gamma-generating matrices, J-inner matrix functions and related extrapolation problems, J. Soviet Math. 52 (1990), 3487-3491; 52 (1990), 3421-3424; 53 (1992), 532-537. MR 1009146 (92i:30034a)
- [Arov2]
- -, Regular J-inner matrix-functions and related continuation problems, Linear Operators in Function Spaces (H. Helson, B. Sz.-Nagy, F. H. Vasilescu, and Gr. Arsene, eds.), Oper. Theory Adv. Appl. vol. 43, Birkhäuser, Basel, 1990, p. 63-87. MR 1090117 (93b:47028)
- [AG]
- D. Z. Arov and L. Z. Grossman, Scattering matrices in the theory of unitary extensions of isometric operators, Math. Nachr. 157 (1992), 105-123. MR 1233051 (94i:47019)
- [AK]
- D. Z. Arov and M. G. Krein, Calculation of entropy functionals and their minima in indeterminate continuation problems, Acta Sci. Math. (Szeged) 45 (1983), 33-50. MR 708770 (84h:30050)
- [Aroc1]
- R. Arocena, Generalized Toeplitz kernels and dilations of intertwining operators, Integral Equations Operator Theory 6 (1983), 759-788. MR 719104 (85i:47008)
- [Aroc2]
- -, Generalized Toeplitz kernels and dilations of intertwining operators, II. The continuous case, Acta Sci. Math. (Szeged) 53 (1989), 123-137. MR 1018681 (91h:47008)
- [BaC]
- M. Bakonyi and T. Constantinescu, Schur's algorithm and several applications, Pitman Res. Notes Math. Ser., vol. 61, Longman, Essex, 1992.
- [Ba1]
- J. A. Ball, Models for noncontractions, J. Math. Anal. Appl. 52 (1975), 235-254. MR 0388135 (52:8972)
- [Ba2]
- -, Nevanlinna-Pick interpolation: Generalizations and applications, Surveys of Some Recent Results in Operator Theory I (J. B. Conway and B. B. Morrel, eds.), Pitman, Boston, 1988, pp. 51-94. MR 958570 (90g:47002)
- [BaC]
- J. A. Ball and N. Cohen, Sensitivity minimization in an norm: parametrization of all suboptimal solutions, Internat. J. Control 46 (1987), 785-816. MR 908751 (88j:93045)
- [BGR]
- J. A. Ball, I. Gohberg, and L. Rodman, Interpolation of rational matrix functions, Birkhäuser-Verlag, Basel, 1990. MR 1083145 (92m:47027)
- [BH]
- J. A. Ball and J. W. Helton, Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic functions: parametrization of the set of all solutions, Integral Equations Operator Theory 9 (1986), 155-203. MR 837514 (87j:30085)
- [BR]
- J. A. Ball and A. C. Ran, Optimal Hankel norm model reductions and Wiener Hopf factorization II: The non-canonical case, Integral Equations Operator Theory 10 (1987), 416-436. MR 883496 (88i:93023)
- [dB]
- L. deBranges, Some Hilbert spaces of analytic functions I, Trans. Amer. Math. Soc. 106 (1963), 445-468. MR 0145335 (26:2866)
- [Br]
- P. Bruinsma, Degenerate interpolation problems for Nevanlinna pairs, Indag. Math. 2 (1991), 179-200. MR 1123360 (92g:47020)
- [BD]
- A. Bultheel and P. Dewilde, On the Adamjan-Arov-Krein approximation, identification and balanced realization of a system, Technical Report TW48, Katholieke Universiteit Leuven, 1980; a short version appears in Proceedings ECTI (1980), 186-191.
- [CP]
- B. C. Chang and J. B. Pearson, Optimal disturbance reduction in linear multivariable systems, IEEE Trans. Automat. Control AC-29 (1984), 880-887. MR 760526 (85m:49074)
- [CS]
- M. Cotlar and C. Sadosky, Toeplitz liftings of Hankel forms, Function Spaces and Applications (M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin, eds.), Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 22-43. MR 942255 (89k:47039)
- [CZ]
- R. F. Curtain and H. J. Zwart, The Nehari problem or the Pritchard-Salmon class of infinite-dimensional linear system: A direct approach, preprint, 1993.
- [DGK]
- Ph. Delsarte, Y. Genin, and Y. Kamp, On the role of the Nevanlinna-Pick problem in circuit and system theory, Internat. J. Circuit Theory Appl. 9 (1981), 177-187. MR 612269 (82d:94052)
- [DD]
- P. Dewilde and H. Dym, Lossless chain scattering matrices and optimum linear prediction: The vector case, Internat. J. Circuit Theory Appl. 9 (1981), 135-175. MR 612268 (82d:94053)
- [DGKF]
- J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, State space solutions to standard and control problems, IEEE Trans. Autom. Control AC-34 (1989), 831-846. MR 1004301 (90m:93055)
- [D1]
- H. Dym, J contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS Reg. Conf. Ser. in Math., vol. 71, Amer. Math. Soc., Providence, RI, 1989. MR 1004239 (90g:47003)
- [D2]
- -, On reproducing kernel spaces, J unitary matrix functions, interpolation and displacement rank, The Gohberg Anniversary Collection II (H. Dym, S. Goldberg, M. A. Kaashoek, and P. Lancaster, eds.), Oper. Theory Adv. Appl., vol. 41, Birkhäuser-Verlag, Basel, 1989, pp. 173-239. MR 1038337 (92f:46024)
- [D3]
- -, On reproducing kernels and the continuous covariance extension problem, Analysis and Partial Differential Equations: A collection of Papers dedicated to Mischa Cotlar (C. Sadosky, ed.), Marcel Dekker, New York, 1990, pp. 427-482. MR 1044802 (92b:42014)
- [D4]
- -, Shifts, realizations and interpolation, redux, Operator Theory and its Applications (A. Feintuch and I. Gohberg, eds.), Oper. Theory Adv. Appl., Birkhäuser-Verlag, Basel, 1994.
- [D5]
- -, On the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares, Comm. Pure Appl. Math. 47 (1994), 1-53. MR 1263129 (95e:47018)
- [DFK]
- V. K. Dubovoj, B. Fritzsche, and B. Kirstein, Marticial version of the classical Schur problem, Teubner, Leipzig, 1992. MR 1152328 (93e:47021)
- [EGL]
- R. L. Ellis, I. Gohberg, and D. C. Lay, Extensions with positive real part, a new version of the abstract band method with applications, Integral Equations Operator Theory 16 (1993), 360-384. MR 1209305 (94a:47024)
- [FFHKP]
- C. Foias, B. Francis, J. W. Helton, H. Kwakernaak, and J. B. Pearson, -control theory (E. Mosca and L. Pandolfi, eds.), Lecture Notes in Math., vol. 1496, Springer-Verlag, Berlin, 1991.
- [FF]
- C. Foias and A. Frazho, The commutant lifting approach to interpolation problems, Birkhäuser-Verlag, Basel, 1990. MR 1120546 (92k:47033)
- [FT]
- C. Foias and A. Tannenbaum, On the four block problem II: The singular system, Integral Equations Operator Theory 11 (1988), 726-767. MR 957193 (89h:47037b)
- [Fr]
- B. A. Francis, A course in control theory, Lecture Notes in Control and Inform. Sci., vol. 88, Springer-Verlag, Berlin, 1987.
- [FrD]
- B. A. Francis and J. C. Doyle, Linear control theory with an optimality criterion, SIAM J. Control Optim. 25 (1987), 815-844. MR 893986 (88e:93032)
- [FrZ]
- B. A. Francis and G. Zames, On -optimal sensitivity theory for SISO feedback systems, IEEE Trans. Automat. Control AC-29 (1984), 9-16. MR 734241 (85g:93030)
- [GK]
- Y. V. Genin and S. Y. Kung, A two-variable approach to the model reduction problem with Hankel norm criterion, IEEE Trans. Circuits and Systems CAS-28 (1981), 912-924. MR 629917 (83b:93013)
- [GeK]
- T. T. Georgiou and P. P. Khargonekar, Spectral factorization and Nevanlinna-Pick interpolation, SIAM J. Control Optim. 25 (1987), 754-766. MR 885197 (88b:93026)
- [GeS]
- T. T. Georgiou and M. C. Smith, Topological approaches to robustness, Lecture Notes in Control and Inform. Sci., vol. 185, Springer-Verlag, New York, 1993, pp. 222-241. MR 1208272 (93k:93038)
- [Glo]
- K. Glover, A tutorial on Hankel norm approximation, From Data to Model (J. C. Willems, ed.), Springer, Berlin, 1989, pp. 26-48.
- [GloD]
- K. Glover and J. C. Doyle, A state space approach to optimal control, Three Decades of Mathematical System Theory, Lecture Notes in Control and Inform. Sci., vol. 135, Springer-Verlag, Berlin, 1989, pp. 179-218. MR 1025791 (90k:93020)
- [GLDKS]
- K. Glover, D. J. N. Limebeer, J. C. Doyle, E. M. Kasenally, and M. G. Safonov, A characterization of all solutions to the four block general distance problem, SIAM J. Control Optim. 29 (1991), 283-324. MR 1092729 (92b:93033)
- [Gr]
- M. Green, controller synthesis by J-lossless coprime factorization, SIAM J. Control Optim. 30 (1992), 522-547. MR 1160141 (93e:93025)
- [GGLD]
- M. Green, K. Glover, D. J. N. Limebeer, and J. C. Doyle, A J-spectral factorization approach to control, SIAM J. Control Optim. 28 (1990), 1350-1371. MR 1075207 (92c:93040)
- [GrL]
- M. Green and D. J. N. Limebeer, Vector interpolation, control and model reduction, Robust of Linear Systems and Nonlinear Control (M. A. Kaashoek, J. H. van Schuppen, and A. C. M. Ran, eds.), Birkhäuser, Boston, 1990, pp. 285-292. MR 1115399 (92g:93046)
- [He1]
- J. W. Helton, Orbit structure of the Möbus transformation semigroup acting on (broadband matching), Topics in Functional Analysis (I. Gohberg and M. Kac, eds.), Academic Press, New York, 1978, pp. 129-157. MR 538017 (81e:46019)
- [He2]
- -, Operator theory, analytic functions, matrices and electrical engineering, CBMS Regional Conference. Ser. in Math., vol. 68, Amer. Math. Soc., Providence, RI, 1987. MR 896034 (89f:47001)
- [HSP]
- G. Herglotz, I. Schur, G. Pick, R. Nevanlinna, and H. Weyl, Ausgewählte Arbeiten zu den Ursprüngen der Schur-Analysis, Teubner, Leipzig, 1991. MR 1162976 (93d:01095)
- [K1]
- V. E. Katsnelson, Methods of J-theory in continuous interpolation problems of analysis, private translation, T. Ando, Sapporo, 1985. MR 777324 (86i:47048)
- [K2]
- -, The problem of the completion of a partial matrix valued function as a classical interpolation problem, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen (V. A. Marchenko, ed.), vol. 56, Vishcha Shkola, Kharkov, 1991, pp. 105-122. MR 1220900 (94g:47018)
- [KKY]
- V. E. Katsnelson, A. Ya. Kheifets, and P. M. Yuditskii, An abstract interpolation problem and the theory of extensions of isometric operators, Operators in Function Spaces and Problems in Function Theory, vol. 146, "Naukova Dumka", Kiev, 1987, pp. 83-96. MR 946502 (90b:47015)
- [Ki1]
- H. Kimura, Directional interpolation approach to optimization and robust stabilization, IEEE Trans. Automat. Control AC-32 (1987), 1085-1093. MR 914841
- [Ki2]
- -, Conjugation interpolation and model matching in , Internat. J. Control 49 (1989), 269-307. MR 981729 (89k:93060)
- [Ki3]
- -, State space approach to the classical interpolation problem and its applications, Three Decades in Mathematical System Theory (H. Nijmeyer and J. M. Schumacher, eds.), Lecture Notes in Control and Inform. Sci., vol. 135, Springer-Verlag, Berlin, 1989, pp. 243-275. MR 1025793 (91b:93124)
- [KL]
- S. Y. King and D. W. Lin, Optimal Hankel-norm model reductions: multivariable systems, IEEE Trans. Automat. Control AC-26 (1981), 832-852. MR 635843 (84a:93017a)
- [KPot]
- I.V. Kovalishina, Analytic theory of a class of interpolation problems, Math. USSR-Izv. 22 (1984), 419-463. MR 703593 (84i:30043)
- [KL1]
- M. G. Krein and H. Langer, On some extension problems which are closely connected with the theory of hermitian operators in a space , III. Indefinite analogues of the Hamburger and Stieltjes moment problems, Beiträge zur Analysis 14 (1979), 25-40; 15 (1980), 27-45.
- [KL2]
- -, On some continuation problems which are closely related to the theory of operators in spaces . IV: Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions, J. Operator Theory 13 (1985), 299-417. MR 776000 (87h:47084)
- [KN]
- M. G. Krein and A. A. Nudelman, The Markov moment problem and extremal problems, Transl. Math. Monographs, vol. 50, Amer. Math. Soc., Providence, RI, 1977. MR 0458081 (56:16284)
- [LA]
- D. J. N. Limebeer and B. D. O. Anderson, An interpolation theory approach to controller degree bounds, Linear Algebra Appl. 89 (1988), 347-386. MR 919123 (88m:93069)
- [LGr]
- D. J. N. Limebeer and M. Green, Parametric interpolation, -control and model reduction, Internat. J. Control 52 (1990), 293-318. MR 1061723 (91i:93045)
- [LK]
- D. J. N. Limebeer and E. M. Kasenelly, Necessary and sufficient conditions for the existence of -optimal controllers; an interpolation approach, Robust Control of Linear Systems and Nonlinear Control (M. A. Kaashoek, J. H. van Schuppen, and A. C. M. Ran, eds.), Birkhäuser, Boston, 1990, pp. 309-316.
- [Nu]
- A. A. Nudelman, On a new problem of moment type, Soviet Math. Doklady 18 (1977), 507-510.
- [Pa]
- L. B. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc. 150 (1970), 529-534. MR 0273449 (42:8327)
- [RR]
- M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford Univ. Press, London, 1986. MR 822228 (87e:47001)
- [Sa]
- D. Sarason, Generalized interpolation in , Trans. Amer. Math. Soc. 127 (1967), 180-203. MR 0208383 (34:8193)
- [SNF1]
- B. Sz.-Nagy and C. Foias, Commutants de certains operateurs, Acta Sci. Math. (Szeged) 29 (1968), 1-17. MR 0242011 (39:3346)
- [SNF2]
- -, Dilation des commutants d'opérateurs, C. R. Acad. Sci. Paris Sér. I Math. 266 (1968), 493-495.
- [SNF3]
- -, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970.
- [Ta]
- A. Tannenbaum, Modified Nevanlinna-Pick interpolation of linear plants with uncertainty in the gain factor, Internat. J. Control 36 (1982), 331-336. MR 674395 (84c:93038)
- [Tad1]
- G. Tadmor, An interpolation problem associated with -optimal design in systems with distributed input lags, Systems Control Lett. 8 (1987), 313-319. MR 884880 (88c:93024)
- [Tad2]
- -, The standard problem and the maximum principle: The general linear case, SIAM J. Control Optim. 31 (1993), 813-846. MR 1227535 (94j:93031)
- [YS]
- D. C. Youla and M. Saito, Interpolation with positive real functions, J. Franklin Inst. 284 (1967), 77-108. MR 0255824 (41:484)
- [Y]
- N. Young, The Nevanlinna-Pick problem for matrix valued functions, J. Operator Theory 15 (1986), 239-265. MR 833210 (87h:30080)
- [Z]
- G. Zames, Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses, IEEE Trans. Automat. Control AC-26 (1981), 301-320. MR 613539 (82d:93017)
Review Information:
Reviewer:
Harry Dym
Journal:
Bull. Amer. Math. Soc.
31 (1994), 125-140
DOI:
https://doi.org/10.1090/S0273-0979-1994-00499-9