Abstract
In Section 1 we recall the setting and solution of the Abstract Interpolation Problem (AIP) from [1]. In Section 2 we rephrase the AIP in terms of unitary scattering systems rather than in terms of unitary colligations. This allows us to give up the orthogonality assumption on the data scales and to formulate a more general setting of the AIP that corresponds to interpolation problems for harmonic functions also. (The original formulation of the AIP corresponded naturally to interpolating analytic functions only.) In Section 3 we give a complete solution to this more general AIP under an additional assumption regarding the data scale ρ0. Solutions are the spectral functions of the feedback coupling with respect to the scale ρ0. In Section 4 we give up the additional assumption of Section 3 regarding the data scale ρ0 and define the scale ρ associated with any feedback coupling by means of the corresponding wave operator and develop the appropriate modification of the results of Section 3. In Section 5 a remark is given on the feedback coupling of the scattering systems. We plan to demonstrate applications of this approach to the General Commutant Lifting problem at another occasion.
Dedicated to Professor Harry Dym on the occasion of his 60th birthday with deep appreciation
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References
V.E. Katsnelson, A.Ya. Kheifets, P.M. Yuditskii, Abstract Interpolation Problem and Isometric Operators Extension Theory, Operators in Functional Spaces and Questions of Function Theory, Kiev (1987) 83–96, Russian. English transi. in Topics in Interpolation Theory (Leipzig,1994), Operator Theory: Advances and Applications, 95(1997) 283–298, Birkhäuser Verlag, Basel.
A.Ya. KheifetsParseval equality in abstract interpolation problem and coupling of open systems (I)Teor. Funk., Funk. Anal. i ikh Prilozhen, 49 (1988) 112–120, Russian. English transl.: J. Sov. Math. 49, 4 (1990) 1114–1120.
A.Ya. KheifetsParseval equality in abstract interpolation problem and coupling of open systems (II)Teor. Funk., Funk Anal. I ikh Prilozhen, 50 (1988) 98–103, Russian. English transi.: J. Sov. Math. 49, 6 (1990) 1307–1310.
A. KheifetsScattering matrices and Parseval equality in Abstract Interpolation problemPh.D. Thesis, Kharkov State University, 1990.
A.Ya. Kheifets, P.M. YuditskiiAn analysis and extension of V.P. Potapov’s approach to interpolation problems with applications to the generalized bi-tangential Schur-Nevanlinna-Pick problem and J-inner-outer factorizationIn Operator Theory: Advances and Applications, 72 (1994) 133–161, Birkhäuser Verlag, Basel.
A. KheifetsAbstract interpolation problem and some applicationsin: Holomorphic Spaces (S. Axler, J. McCarthy, D. Sarason editors), MSRI Publications, 33 (1998) 351–381, Cambridge University Press.
S.S. Boiko, V.K. Dubovoy, A.Ya. KheifetsMeasure Schur complements and spectral functions of unitary operators with respect to different scalesin D. Alpay and V. Vinikov (editors), Operator Theory, System Theory and Related Topics (The Moshe Livšic Anniversary Volume), Operator Theory: Advances and Applications, 123(2001) 89–138, Birkhäuser, Basel.
A.Ya. KheifetsHamburger moment problem: Parseval equality and Arov-singularityJournal of Functional Analysis, 141, 2 (1996) 374–420.
J. Ball, T. TrentThe abstract interpolation problem and commutant lifting: coordinate-free approachOperator Theory: Advances and Applications, 115 (2000) 51–83, Birkhäuser, Basel.
B.Sz.-Nagy, C. FoiasHarmonic analysis of operators in Hilbert space.North-Holland, Amsterdam, 1970.
D.Z. Arov, L.Z. GrossmanScattering matrices in the extension theory of isomet-ric operatorsSoviet Math. Dokl., 27 (1983) 573–578.
A. KheifetsParameterization of solutions of the Nehari problem and nonorthogonal dynamicsOperator Theory: Advances and Applications, 115 (2000) 213–233, Birkhäuser, Basel.
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Kheifets, A. (2002). Abstract Interpolation Scheme for Harmonic Functions. In: Alpay, D., Vinnikov, V., Gohberg, I. (eds) Interpolation Theory, Systems Theory and Related Topics. Operator Theory: Advances and Applications, vol 134. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8215-6_13
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DOI: https://doi.org/10.1007/978-3-0348-8215-6_13
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