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Book Review
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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.
- [3]
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 (58 #30446)
- [4]
V.
M. Adamjan, D.
Z. Arov, and M.
G. Kreĭn, Bounded operators which commute with a
𝐶₀₀ class contraction whose rank of nonunitarity is
one, Funkcional. Anal. i Priložen 3 (1969),
no. 3, 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,
Basel, 1992, pp. 1–51. MR
1173917
- [7]
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
- [8]
J.
A. Ball, I.
Gohberg, and M.
A. Kaashoek, Time-varying systems: Nevanlinna-Pick interpolation
and sensitivity minimization, and signal processing, I (Kobe, 1991)
Mita, Tokyo, 1992, pp. 53–58. MR 1197893
(93m:93027)
- [9]
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
(88g:47027), http://dx.doi.org/10.1007/BF01195035
- [10]
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 (88c:47030), http://dx.doi.org/10.1016/0024-3795(87)90297-7
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H.
Bart, I.
Gohberg, and M.
A. Kaashoek, The state space method in problems of analysis,
applied mathematics (ICIAM 87) (Paris, 1987) CWI Tract, vol. 36,
Math. Centrum Centrum Wisk. Inform., Amsterdam, 1987, pp. 1–16.
MR
905689
- [12]
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
- [13]
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, 1989. MR 1004239
(90g:47003)
- [14]
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
(92k:47033)
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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 0409824
(53 #13576)
- [16]
Keith
Glover, All optimal Hankel-norm approximations of linear
multivariable systems and their 𝐿^{∞}-error bounds,
Internat. J. Control 39 (1984), no. 6,
1115–1193. MR 748558
(86a:93029), http://dx.doi.org/10.1080/00207178408933239
- [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 (Rehovot, 1983) Oper. Theory Adv. Appl., vol. 12,
Birkhäuser, Basel, 1984, pp. 241–275. MR 761363
(86k:93035)
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I.
Gohberg, P.
Lancaster, and L.
Rodman, Spectral analysis of matrix polynomials. I. Canonical forms
and divisors, Linear Algebra and Appl. 20 (1978),
no. 1, 1–44. MR 0463199
(57 #3155)
- [19]
I.
Gohberg, P.
Lancaster, and L.
Rodman, Spectral analysis of matrix polynomials. II. The resolvent
form and spectral divisors, Linear Algebra and Appl.
21 (1978), no. 1, 65–88. MR 0498634
(58 #16720)
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I.
Gohberg, P.
Lancaster, and L.
Rodman, Representations and divisibility of operator
polynomials, Canad. J. Math. 30 (1978), no. 5,
1045–1069. MR 506259
(80a:47024), http://dx.doi.org/10.4153/CJM-1978-088-2
- [21]
I.
Gohberg, P.
Lancaster, and L.
Rodman, Matrix polynomials, Academic Press Inc. [Harcourt
Brace Jovanovich Publishers], New York, 1982. Computer Science and Applied
Mathematics. 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.
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, 1987. MR 896034
(89f:47001)
- [24]
Thomas
Kailath, Linear systems, Prentice-Hall Inc., Englewood Cliffs,
N.J., 1980. Prentice-Hall Information and System Sciences Series. 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.
L. Falb, and M.
A. Arbib, Topics in mathematical system theory, McGraw-Hill
Book Co., New York, 1969. MR 0255260
(40 #8465)
- [27]
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
(89f:00030)
- [28]
R. Nevanlinna, Über beschränkte analytische Funktionen, Ann. Acad. Sci. Fenn. Ser. A I Math. 32 (1929), no. 7.
- [29]
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
(87e:47001)
- [30]
Donald
Sarason, Generalized interpolation in
𝐻^{∞}, Trans. Amer. Math.
Soc. 127 (1967),
179–203. MR 0208383
(34 #8193), http://dx.doi.org/10.1090/S0002-9947-1967-0208383-8
- [31]
Béla
Sz.-Nagy and Ciprian
Foiaș, Harmonic analysis of operators on Hilbert space,
Translated from the French and revised, North-Holland Publishing Co.,
Amsterdam, 1970. MR 0275190
(43 #947)
- [32]
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 0087062
(19,296e)
- [33]
Béla
Sz.-Nagy and Adam
Korányi, Operatortheoretische Behandlung und
Verallgemeinerung eines Problemkreises in der komplexen
Funktionentheorie, Acta Math. 100 (1958),
171–202 (German). MR 0130577
(24 #A437)
- [34]
H.
J. Woerdeman, Matrix and operator extensions, CWI Tract,
vol. 68, Stichting Mathematisch Centrum Centrum voor Wiskunde en
Informatica, Amsterdam, 1989. MR 1037264
(91d:47001)
- [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:
http://dx.doi.org/10.1090/S0273-0979-1993-00386-0
PII:
S 0273-0979(1993)00386-0
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