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Regular and singular J-inner matrix-functions and the corresponding interpolation problems. (English. Russian original) Zbl 0661.30029
Funct. Anal. Appl. 22, No. 1, 46-48 (1988); translation from Funkts. Anal. Prilozh. 22, No. 1, 57-59 (1988).
For J a given signature matrix \((J=J^*=J^{-1})\), a matrix function W meromorphic on E (when E is either the unit disk D or the upper half plane \(\pi)\) is said to be J-inner if W has J-contractive values on E and J-unitary a.e. existing boundary values on the boundary of E. Such a function W is said to be a singular J-inner matrix function if both W and \(W^{-1}\) can be expressed as the product of a bounded (and analytic on E) matrix function with the inverse of a bounded scalar outer function.
A J-inner function W is said to be regular if it has no nonconstant right singular divisor in the class of J-inner matrix functions. When \(J=I\), all inner functions are regular, but for a general J, even rational nonconstant singular J-inner functions exist; such functions necessarily have all poles on the boundary of E and are called Brune sections in circuit theory.
J-inner functions arise naturally as the provider of a linear fractional parametrization of all solutions of a Nevanlinna-Pick-type interpolation problem; singular J-inner functions arise when the interpolation problem involves prescribed values for the function and its angular derivative on the boundary of E. The present paper obtains an explicit function theoretic characterization of singular J-inner functions, presents some elementary properties of the singular-regular factorization for J-inner functions, and solves the converse problem for finding the data for an interpolation problem in terms of a given J-inner function which parametrizes the solution set. All these results are for general (not necessarily rational) J-inner matrix functions.
Reviewer: J.A.Ball

30E05 Moment problems and interpolation problems in the complex plane
46E40 Spaces of vector- and operator-valued functions
47B50 Linear operators on spaces with an indefinite metric
Full Text: DOI
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