Book Review
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1568031
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Book Information:
Author:
I. Gohberg J.~A. Ball, and L. Rodman
Title:
Interpolation of rational matrix functions
Additional book information:
Operator Theory: Advances and Applications, vol. 45, Birkh\"aus\-er Verlag, Basel, 1990, 605 pp., US$129.00. ISBN 3-7643-2476-7.
[1] V. M. Adamjan, D. Z. Arov, and M. G. Krein, Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem, Math. USSR-Sb. 15 (1971), 31-73.
[2] -, Infinite block Hankel matrices and their connections with the interpolation problem, Akad. Nauk Armenia SSR Isv. Mat. 6 (1971); English transl. in Amer. Math. Soc. Transl. Ser. 2 111, (1978), Amer. Math. Soc., Providence, RI.
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
V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Bounded operators which commute with a $C_{00}$ class contraction whose rank of nonunitarity is one, Funkcional. Anal. i Priložen 3 (1969), no. 3, 86–87 (Russian). MR 0261375
[5] -, Infinite Hankel block matrices and related continuation problems, Izv. Akad. Nauk Armyan. SSSR Ser. Mat. 6 (1978), 87-112. (Russian)
J. A. Ball, I. Gohberg, and M. A. Kaashoek, Nevanlinna-Pick interpolation for time-varying input-output maps: the discrete case, Time-variant systems and interpolation, Oper. Theory Adv. Appl., vol. 56, Birkhäuser, Basel, 1992, pp. 1–51. MR 1173917
J. A. Ball, I. Gohberg, and M. A. Kaashoek, Nevanlinna-Pick interpolation for time-varying input-output maps: the continuous time case, Time-variant systems and interpolation, Oper. Theory Adv. Appl., vol. 56, Birkhäuser, Basel, 1992, pp. 52–89. MR 1173918
J. A. Ball, I. Gohberg, and M. A. Kaashoek, Time-varying systems: Nevanlinna-Pick interpolation and sensitivity minimization, Recent advances in mathematical theory of systems, control, networks and signal processing, I (Kobe, 1991) Mita, Tokyo, 1992, pp. 53–58. MR 1197893
Joseph A. Ball and André C. M. Ran, Local inverse spectral problems for rational matrix functions, Integral Equations Operator Theory 10 (1987), no. 3, 349–415. MR 883495, DOI 10.1007/BF01195035
Joseph A. Ball and André C. M. Ran, Global inverse spectral problems for rational matrix functions, Linear Algebra Appl. 86 (1987), 237–282. MR 870942, DOI 10.1016/0024-3795(87)90297-7
H. Bart, I. Gohberg, and M. A. Kaashoek, The state space method in problems of analysis, Proceedings of the first international conference on industrial and applied mathematics (ICIAM 87) (Paris, 1987) CWI Tract, vol. 36, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1987, pp. 1–16. MR 905689
Patrick Dewilde and Harry Dym, Interpolation for upper triangular operators, Time-variant systems and interpolation, Oper. Theory Adv. Appl., vol. 56, Birkhäuser, Basel, 1992, pp. 153–260. MR 1173921
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
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
I. P. Fedčina, The tangential Nevanlinna-Pick problem with multiple points, Akad. Nauk Armjan. SSR Dokl. 61 (1975), no. 4, 214–218 (Russian, with Armenian summary). MR 409824
Keith Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $L^{\infty }$-error bounds, Internat. J. Control 39 (1984), no. 6, 1115–1193. MR 748558, DOI 10.1080/00207178408933239
I. Gohberg, M. A. Kaashoek, L. Lerer, and L. Rodman, Minimal divisors of rational matrix functions with prescribed zero and pole structure, Topics in operator theory systems and networks (Rehovot, 1983) Oper. Theory Adv. Appl., vol. 12, Birkhäuser, Basel, 1984, pp. 241–275. MR 761363
I. Gohberg, P. Lancaster, and L. Rodman, Spectral analysis of matrix polynomials. I. Canonical forms and divisors, Linear Algebra Appl. 20 (1978), no. 1, 1–44. MR 463199, DOI 10.1016/0024-3795(78)90026-5
I. Gohberg, P. Lancaster, and L. Rodman, Spectral analysis of matrix polynomials. II. The resolvent form and spectral divisors, Linear Algebra Appl. 21 (1978), no. 1, 65–88. MR 498634, DOI 10.1016/0024-3795(87)90201-1
I. Gohberg, P. Lancaster, and L. Rodman, Representations and divisibility of operator polynomials, Canadian J. Math. 30 (1978), no. 5, 1045–1069. MR 506259, DOI 10.4153/CJM-1978-088-2
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
[22] I. C. Gohberg and E. I. Sigal, On operator generalizations of the logarithmic residue theorem and the theorem of Rouché, Math. USSR-Sb. 13 (1971), 603-625.
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
Thomas Kailath, Linear systems, Prentice-Hall Information and System Sciences Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. MR 569473
[25] M. V. Keldysh, On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations, DAN SSSR 77 (1951), 11-14. (Russian)
R. E. Kalman, P. L. Falb, and M. A. Arbib, Topics in mathematical system theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1969. MR 0255260
Ben Silver (ed.), American Mathematical Society Translations. Series 2. Vol. 138, American Mathematical Society Translations, Series 2, vol. 138, American Mathematical Society, Providence, RI, 1988. Seven papers translated from the Russian; Translation by H. H. McFaden. MR 940259, DOI 10.1090/trans2/138
[28] R. Nevanlinna, Über beschränkte analytische Funktionen, Ann. Acad. Sci. Fenn. Ser. A I Math. 32 (1929), no. 7.
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, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
B. Sz.-Nagy and A. Korányi, Relations d’un problème de Nevanlinna et Pick avec la théorie des opérateurs de l’espace hilbertien, Acta Math. Acad. Sci. Hungar. 7 (1956), 295–303 (French, with Russian summary). MR 87062, DOI 10.1007/BF02020526
Béla Sz.-Nagy and Adam Korányi, Operatortheoretische Behandlung und Verallgemeinerung eines Problemkreises in der komplexen Funktionentheorie, Acta Math. 100 (1958), 171–202 (German). MR 130577, DOI 10.1007/BF02559538
H. J. Woerdeman, Matrix and operator extensions, CWI Tract, vol. 68, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 1037264
- [1]
- V. M. Adamjan, D. Z. Arov, and M. G. Krein, Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem, Math. USSR-Sb. 15 (1971), 31-73.
- [2]
- -, Infinite block Hankel matrices and their connections with the interpolation problem, Akad. Nauk Armenia SSR Isv. Mat. 6 (1971); English transl. in Amer. Math. Soc. Transl. Ser. 2 111, (1978), Amer. Math. Soc., Providence, RI.
- [3]
- -, Infinite Hankel and generalized Carathéodory-Fejér and I. Schur problems, Funktsional. Anal, i Prilozhen 2 (1968), 1-17. (Russian) MR 0636333 (58:30446)
- [4]
- -, Bounded operators that commute with a contraction of class
of unit rank of nonunitary, Funktsional. Anal, i Prilozhen 3 (1969), 86-87. (Russian) MR 0261375 (41:5990)
- [5]
- -, Infinite Hankel block matrices and related continuation problems, Izv. Akad. Nauk Armyan. SSSR Ser. Mat. 6 (1978), 87-112. (Russian)
- [6]
- J. A. Ball, I. Gohberg, and M. A. Kaashoek, Nevanlinna-Pick interpolation for time-varying input-output maps: the discrete case, Time-variant Systems and Interpolation, Oper. Theory: Adv. Appl., vol. 56, Birkhäuser Verlag, Basel, 1992, pp. 1-51. MR 1173917
- [7]
- -, Nevanlinna-Pick interpolation for time-varying input-output maps: the continuous time case, Time-variant Systems and Interpolation, Oper. Theory: Adv. Appl., vol. 56, Birkhäuser Verlag, Basel, 1992, pp. 52-89. MR 1173918
- [8]
- -, Time-varying systems: Nevanlinna-Pick interpolation and sensitivity minimization, Recent Advances in Mathematical Theory of Systems, Control, Networks and Signal Processing I (H. Kimura and S. Kodama, eds.), Proceedings of the International Symposium MTNS-91, Mita Press, Tokyo, 1992, pp. 53-58. MR 1197893 (93m:93027)
- [9]
- J. A. Ball and A. C. M. Ran, Local inverse spectral problems for rational matrix functions, Integral Equations Operator Theory 10 (1987), 349-415. MR 883495 (88g:47027)
- [10]
- -, Global inverse spectral problems for rational matrix functions, Linear Algebra Appl. 86 (1987), 237-282. MR 870942 (88c:47030)
- [11]
- H. Bart, I. Gohberg, and M. A. Kaashoek, The state space method in problems of analysis, Proceedings First International Conference on Industrial and Applied Mathematics, Contributions from the Netherlands, CWI, Amsterdam, 1987, pp. 1-16. MR 905689
- [12]
- P. Dewilde and H. Dym, Interpolation of upper triangular operators, Time-variant Systems and Interpolation, Oper. Theory: Adv. Appl., vol. 56, Birkhäuser Verlag, Basel, 1992, pp. 153-260. MR 1173921
- [13]
- H. Dym, J contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS Regional Conf. Ser. in Math., vol. 71, Amer. Math. Soc., Providence, RI, 1989. MR 1004239 (90g:47003)
- [14]
- C. Foias and A. E. Frazho, The commutant lifting approach to interpolation problems, Oper. Theory: Adv. Appl., vol. 44, Birkhäuser Verlag, Basel, 1990. MR 1120546 (92k:47033)
- [15]
- I. P. Fedchina, Nevanlinna-Pick problem with multiple points, Doklady Akad. Nauk Arm. SSR 61 (1975), 214-218. (Russian) MR 0409824 (53:13576)
- [16]
- K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their
error bounds, Internat. J. Control 39 (1984), 1115-1193. MR 748558 (86a:93029)
- [17]
- I. Gohberg, M. A. Kaashoek, L. Lerer, and L. Rodman, Minimal divisors of rational matrix functions with prescribed zero and pole structure, Topics in Operator Theory, Systems and Networks (H. Dym and I. Gohberg, eds.), Oper. Theory: Adv. Appl., vol. 12, Birkhäuser Verlag, Basel, 1984, pp. 241-275. MR 761363 (86k:93035)
- [18]
- I. Gohberg, P. Lancaster, and L. Rodman, Spectral analysis of matrix polynomials, I. Canonical forms and divisors, Linear Algebra Appl. 20 (1978), 1-44. MR 0463199 (57:3155)
- [19]
- -, Spectral analysis of matrix polynomials, II. The resolvent form and spectral divisors, Linear Algebra Appl. 21 (1978), 65-88. MR 0498634 (58:16720)
- [20]
- -, Representation and divisibility of operator polynomials, Canad. Math. J. 30 (1978), 1045-1069. MR 506259 (80a:47024)
- [21]
- -, Matrix polynomials, Academic Press, New York, 1982. MR 662418 (84c:15012)
- [22]
- I. C. Gohberg and E. I. Sigal, On operator generalizations of the logarithmic residue theorem and the theorem of Rouché, Math. USSR-Sb. 13 (1971), 603-625.
- [23]
- J. W. Helton, Operator theory, analytic functions, matrices and electrical engineering, CBMS Regional Conf. Ser. in Math., vol. 68, Amer. Math. Soc., Providence, RI, 1987. MR 896034 (89f:47001)
- [24]
- T. Kailath, Linear systems, Prentice Hall, Englewood Cliffs, NJ, 1980. MR 569473 (82a:93001)
- [25]
- M. V. Keldysh, On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations, DAN SSSR 77 (1951), 11-14. (Russian)
- [26]
- R. E. Kalman, P. Falb, and M. Arbib, Topics in mathematical system theory, McGraw-Hill, New York, 1969. MR 0255260 (40:8465)
- [27]
- I. V. Kovalishna and V. P. Potapov, Seven papers translated from the Russian, Amer. Math. Soc. Transl. Ser. 2, vol. 138, Amer. Math. Soc., Providence, RI, 1988. MR 940259 (89f:00030)
- [28]
- R. Nevanlinna, Über beschränkte analytische Funktionen, Ann. Acad. Sci. Fenn. Ser. A I Math. 32 (1929), no. 7.
- [29]
- M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford Univ. Press, Oxford, 1985. MR 822228 (87e:47001)
- [30]
- D. Sarason, Generalized interpolation in
, Trans. Amer. Math. Soc. 127 (1967), 179-203. MR 0208383 (34:8193)
- [31]
- B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, American Elsevier, New York, 1970. MR 0275190 (43:947)
- [32]
- B. Sz.-Nagy and A. Koranyi, Relations d'une problème de Nevanlinna et Pick avec la théorie des opérateurs de l'espace hilbertien, Acta Math. Sci. Hungar. 7 (1956), 295-302. MR 0087062 (19:296e)
- [33]
- -, Operator theoretische Behandlung und Veralgemeinerung eines Problemkreises in der komplexen Funktionentheorie, Acta Math. 100 (1958), 171-202. MR 0130577 (24:A437)
- [34]
- H. J. Woerdeman, Matrix and operator extensions, CWI Tract 68, Centre for Mathematics and Computer Science, Amsterdam, 1989. MR 1037264 (91d:47001)
Review Information:
Reviewer:
M. A. Kaashoek
Journal:
Bull. Amer. Math. Soc.
28 (1993), 426-434
DOI:
https://doi.org/10.1090/S0273-0979-1993-00386-0