Abstract
Interpolation problems play a significant role both in applied and theoretical investigations. Classical interpolation problems include those of Nevanlinna-Pick, Caratheodory, Schur, and Hamburger, as well as the problem of trigonometric moments.
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References
V.M. Adamjan, D.Z. Arov, M.G. Krein. Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem, Math. USSR-Sb. 15, 31–73, 1971.
M.G. Krein, H. Langer. Uber einige Fortsetzung probleme. Math. Nachr. 77, 187–236, 1977.
A.A. Nudel’man, On a new problem of moment type. Dokl. Akad. Nauk SSSR 233 (1977), 792–795
Soviet Math. Dokl 18 (1977), 507–510.
I.V. Kovalishina, V.P. Potapov, Indefinite metric in the Nevanlinna-Pick problem. Dokl. Akad. Nauk Armyan. SSR, Ser. Mat. ,59 (1974), 17–22
(Translation) Collected Papers of V.P. Potapov ,p. p. 33–40, Hokkaido Univ., 1982, Sapporo.
J-expansive matrix-functions in the Caratheodory problem, Dokl. Akad. Nauk Armyan. SSR, Ser. Mat ,59 (1974), 129–135.
V.E. Katsnel’son, Methods of J-theory in continuous interpolation problems of analysis. Part 1. Hokkaido Univ. 1985, Sapporo.
I.S. Iohvidov, M.G. Krein, H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric. Akademic-Verlag Berlin, Band 9, 1982.
C. Foias, A. Frazho, The Commutant Lifting Approach to Interpolation Problems ,Birkhäuser-Verlag, 1990.
L. de Branges, Hilbert Spaces of Entire Functions ,1968, Prentice-Hall.-326 p.
J. Ball, I. Gohberg, L. Rodman, Interpolation of Rational Matrix Functions. Birkhäuser-Verlag, 1990.
P. Dewilde, A Course on the Algebraic Schur and Nevanlinna-Pick Interpolation Problems. Lectures and Tutorials Presented at the International Workshop on Algorithums and Parallel VLSI Architectures ,Abbaye des Premontres, Pont-a-Mous-son, France, June 10–16, 1990.
H. Nijmeijer, J.M. Schumacher (ed.), Three Decades of Mathematical System Theory ,Springer-Verlag, 1989.-562 p.
V.P. Potapov, The multiplicative structure of J-contractive matrix functions. Trudy Moskov. Mat Obshch. ,4 (1955), 125–236
(Translation) Amer. Math. Soc. Transl. (2) 15 (1960), 131–243.
A.V. Efimov, V.P. Potapov. J-expanding matrix functions and their role in the analytical theory of electrical circuits. Uspehi Mat. Nauk ,28 (1973), no 1 (169) 65–130
(Translation) Russian Math. Survey ,28:1 (1973), 69–140.
V.P. Potapov, General theorems on the structure and splitting-off of elementary factors of analytic matrix-functions. Dokl. Akad. Nauk. Armyan. SSR, Ser. Mat ,48 (1969), 257–262
(Translation) Collected papers of V.P. Potapov ,pp. 23–32, Hokkaido Univ., 1982, Sapporo.
V.P. Potapov, Fundamental facts of the theory of J-contractive matrix-functions. Proc. All Union Conf. on Theory of Functions ,pp. 1979–181, 1971, Khar’kov.
I.V. Kovalishina, V.P. Potapov. The radii of Weyl disc in the tangential Nevanlinna-Pick problem, in Theory of Operators in Function Spaces and its Applications ,pp. 25–49, Naukova Dumka, 1981, Kiev
(Translation) Collected Papers of V.P. Potapov ,pp. 67–99, Hokkaido Univ., 1982, Sapporo.
I.V. Kovalishina, V.P. Potapov. Integral representation of Hermitian positive functions. Deposited to VINITI, no. 2984-81, 1981
V.K. Dubovoi, Indefinite metric in the interpolation Schur problem for analitic functions. Teor Funktsii Funktsional Anal. i Prilozhen. ,I-37 (1982), 14–26
L.A. Sakhnovich, On similarity of linear operators. Siberian Math. J. 13 (1972), 604–515.
T.S. Ivanchenko, L.A. Sakhnovich, Operator approach to the investgation of interpolation problems. Deposited to Ukr. NIINTI, no. 701, Uk-85, 1985.
T.S. Ivanchenko, L.A. Sakhnovich, Operator identities in the theory of interpolation problems. Izv. Akad. Nauk Armyan. SSR. Ser. Mat ,XXII, 3 (1987), 298–308.
T.S. Ivanchenko, L.A. Sakhnovich, Operator approach to Potapov scheme. Ukrain Math. J. 39 no. 5 (1987), 573–578.
A.A. Nudel’man, P.A. Shvartsman, On the existence of solutions of certain operator inequalities, Siberian Math J. 16 (1975) 431–439. Plenum Publishing Co., New York.
L.A. Sakhnovich, Equations with a difference kernel on a finite interval. Russian Math. Surv. 35:4 (1980) 81–152.
L.A. Sakhnovich, Factorization problems and operator identities. Russian Math. Surv. 41:1 (1986) 1–64.
M.G. Krein, Sur le probleme du prolongement des fonctions hermitiennes positivees et continues. Dokl. Akad. Nauk SSSR ,26 (1940), 17–22.
S.A. Orlov, Nested matrix disks, analytically depending on a parameter, and theorems on invariance of ranks of radii of limiting disks. Izv. Akad. Nauk SSSR, Ser. Mat ,40 (1976), 593–644
(Translation) Math USSR-Izv. ,10 (1976), 565–613.
A.L. Sakhnovich, On the extension of Toeplitz matrices and their continuous analogues. Dissertation, Khar’kov (1982).
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Ivanchenko, T.S., Sakhnovich, L.A. (1994). An Operator Approach to the Potapov Scheme for the Solution of Interpolation Problems. In: Gohberg, I., Sakhnovich, L.A. (eds) Matrix and Operator Valued Functions. Operator Theory Advances and Applications, vol 72. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8532-4_4
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DOI: https://doi.org/10.1007/978-3-0348-8532-4_4
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