Abstract Interpolation Scheme for Harmonic Functions
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In Section 1 we recall the setting and solution of the Abstract Interpolation Problem (AIP) from . 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.
KeywordsHarmonic Function Spectral Function Wave Operator Unitary Extension Scattering System
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