The Bitangential Inverse Input Impedance Problem for Canonical Systems,I: Weyl-Titchmarsh Classification,Existence and Uniqueness |
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Authors: | Email author" target="_blank">Damir?Z?ArovEmail author Harry?Dym |
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Institution: | (1) Department of Mathematics, South-Ukranian Pedagogical University, 65020 Odessa, Ukraine;(2) Department of Mathematics, The Weizmann Institute of Science, 76100 Rehovot, Israel |
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Abstract: | The inverse input impedance problem is investigated in the class
of canonical integral systems with matrizants that are strongly regular J-inner
matrix valued functions in the sense introduced in ArD1]. The set of
solutions for a problem with a given input impedance matrix (i.e., Weyl-
Titchmarsh function) is parameterized by chains of associated pairs of entire
inner p × p matrix valued functions. In our considerations the given data for
the inverse bitangential input impedance problem is such a chain and an input
impedance matrix, i.e., a p × p matrix valued function in the Carathéodory
class. Existence and uniqueness theorems for the solution of this problem are
obtained by consideration of a corresponding family of generalized bitangential
Carathéodory interpolation problems. The connection between the inverse
bitangential input scattering problem that was studied in ArD4] and the bitangential
input impedance problem is also exploited. The successive sections
deal with: 1. The introduction, 2. Domains of linear fractional transformations,
3. Associated pairs of the first and second kind, 4. Matrix balls, 5. The
classification of canonical systems via the limit ball, 6. The Weyl-Titchmarsh
characterization of the input impedance, 7. Applications of interpolation to
the bitangential inverse input impedance problem. Formulas for recovering
the underlying canonical integral systems, examples and related results on the
inverse bitangential spectral problem will be presented in subsequent publications.D. Z. Arov thanks the Weizmann Institute of Science for hospitality and support, partially as a
Varon Visiting Professor and partially through the Minerva Foundation. H. Dym thanks Renee
and Jay Weiss for endowing the chair which supports his research and the Minerva Foundation. |
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Keywords: | Primary: 34A55 34B20 30E05 Secondary: 46E22 47B32 |
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